## Infinity and the Mind

The Science and Philosophy of the Infinite

RUDY RUCKER

Copyright © 1982, 1995, 2005, 2019 by Rudy Rucker

*This is a free online webpage browsing edition. Copyright (C) Rudy Rucker 2019.
Purchase paperback and ebook editions from*

Princeton University Press or Amazon or B&N

Originally published by Birkhäuser in 1982.

2nd edition Princeton Science Library, 1995

3rd edition Princeton Science Library, 2005

4th edition Princeton Science Library, 2019, with a new preface, not shown here.

## TABLE OF CONTENTS

## PREFACE TO THE 2005 EDITION

So here I am writing a third preface for *Infinity and the Mind.* Though I keep hoping to surpass it, this early work may turn out to be my most popular. *Habent sua fata libelli*—books have their own destinies.

I’m grateful to my publishers, and to the readers who’ve contacted me over the years. Infinity is important and interesting, and learning about it can indeed make a difference in one’s life.

I’ll take this opportunity to expand upon my text with three notes touching on, respectively, cosmology, computer science, and set theory.

* * *

In the first of the puzzles and paradoxes at the end of Chapter One, I posed the question of whether an infinite universe would be likely to have planets in every way identical to our own. And in my answer on page 295, I minimized this likelihood.

But now I feel that I was underestimating the raw power of infinity. If our universe is infinite, then it could very well be that somewhere far from Earth, a person exactly like you is reading this precise sentence.

The question turns out to be less hypothetical than I formerly supposed, for nowadays some cosmologists believe that our space is truly infinite, with an endless number of stars and planets. The initial Big Bang singularity is to have happened not at any single point, but across infinite space.

If the old image of the Big Bang was of a white dot appearing in a plane, the new image is of an entire endless plane becoming suddenly illuminated in every part. You might visualize a sheet of light settling down upon the plane; indeed, one current model views the universe as pair of parallel spaces that oscillate back and forth, creating a Big Bang each time they pass through each other.

Regarding the question of whether an infinite universe would contain other worlds just like Earth, I recently saw some interesting number play by Max Tegmark in his article “Parallel Universes” (*Scientific American*, May 2003, pp. 41–51). Assuming that the space-filling Big Bang happened 14 billion years ago, the region of the endless universe presently visible to us is a sphere with a diameter of some 1027, or 1 octillion, meters. (I should mention that the standard name for a number of the form 10((*κ* + 1)·3) has the general form *κ*-illion.) This octillion-meter-wide sphere, which is called a Hubble volume, contains those objects that are close enough so that light from them has had time since the Big Bang to reach us.

Suppose we make the unexceptionable assumption that our Hubble volume has an average temperature of less than 100 million degrees centigrade (the sun’s surface is a mere 5000 degrees centigrade). In this case, according to Tegmark, the Hubble volume has room for some 10118 protons. To get a handle on this number it’s useful to use the number *googol*, which is written as a one followed by a hundred zeroes:

10118 = 10(18 + 100) = 1018 · 10100 = 10((5+1) · 3) · l0100 = quintillion googol.

Now we can wonder how many distinct possible Hubble-volume-sized regions there could be. Let’s think of the Hubble volume as a jungle-gym grid with a quintillion googol slots. One can specify an arbitrary random visible universe by deciding what to put in each slot—one might leave a slot empty, put a proton or neutron in there, or perhaps stick in an electron or some other kind of particle. To keep things reasonably simple, let’s suppose we have ten alternate ways to fill each of the quintillion googol proton-sized slots. In that case, the number of possible ways to populate a Hubble volume with matter consists of choosing among ten options a quintillion googol times in a row, which is ten to the quintillion googol power. In describing this number, it will be useful to use googol’s big brother, the *googolplex*, which is 10googol, that is, the number one followed by googol zeroes. (See p. 97 for more about googol and googolplex.)

Since it’s rather hard to typeset double exponents, I’ll use the ^ symbol to stand for the second level of exponentiation.

10(quintillion googol) = 10(10^118) = 10((10^100) · (10^18)) = (10(10^100))(10^18) = googolplexquintillion.

So now we know that there are at most googolplex-to-the-quintillion possible versions of how our visible universe could appear. A large number, yes, but if our universe is truly infinite, there will be an infinite number of possible Hubble volumes besides ours, and it seems likely that one of them could be an exact match for our own.

How far off might the first copy of our visible universe be? One idea might be to set out in a straight line and whip through the first googolplex-to-the-quintillion Hubble volumes. Just for fun, let’s give this distance a made-up name: one *striiide.* Given that a Hubble volume has a diameter of an octillion meters, a striiide is an octillion googolplex-to-the-quintillion meters. Would traveling this far guarantee a hit? Not quite.

A little calculating of probabilities indicates that if I travel one striiide, I have a 63% chance of encountering a Hubble volume exactly like the one I started from (the precise probability is very close to 1 – 1/*e*, where *e* is the base of the natural logarithm). But as I travel more striiides, the odds go up, and after ten striiides, my chances of having found a visible universe exactly like ours is better than 99.99%.

With all this said, there’s still no *certainty* that a supertraveler could find another Hubble volume just like the one right here. Consider for instance an infinite set of integers which has only *one* odd member, the number 3. Someone who starts at 3 and looks for another odd number is going to be disappointed:

{2, 3, 4, 6, 8, 10, 12, … 2*n*, …}.

But realistically there’s no reason to suppose that our particular Hubble volume is unique and special. So, yes, if the universe really is infinite, then there probably are other people exactly like us somewhere out there. It’s an odd thought, somehow liberating. Even if I goof up this preface, some other Rudy will get it right. So why worry?

* * *

I need to make a correction to the section “A Technical Note on Man-Machine Equivalence” on pages 292–294. (I discovered the problem in conversations with the philosophers Leon Horsten and Mark van Atten at the University of Leuven in 2002. I was there as a grateful guest of the Brussels-based VLAC, or Flemish Academic Centre for Science and the Arts.)

I hold with the conjecture that it is in principle possible to bring into existence computing machines that think like humans do. The point of my technical note was to defend my belief against J. Anthony Lucas’s classic argument that Gödel’s Second Incompleteness Theorem rules out man-machine equivalence, an argument which was revived and popularized by physicist-author Roger Penrose in the 1990s.

Unfortunately my note has a mistake in it even though, in the second preface to *Infinity and the Mind*, I claimed that my note was “definitive.” Oh well. I’m another Rudy by now, and I’m here to set things right.

Suppose *h* is an integer that codes the program for a device *M _{h}*, whose output is very much like a person’s. Lucas and Penrose want to say that (1) after hanging around with

*M*for awhile, any reasonable person will feel like asserting Tr(

_{h}*h*), a sentence which says something like, “If I base a machine

*M*on the algorithm coded by

_{h}*h*I’ll get a machine that only outputs true sentences about mathematics” and (2) having perceived the truth of Tr(

*h*), any reasonable person will also feel like asserting Con(

*h*), a sentence which says something like, “If I base a machine

*M*on the algorithm coded by

_{h}*h*I’ll get a machine that never generates any mathematical contradictions.” But Gödel’s Second Incompleteness Theorem shows that

*M*can’t prove Con(

_{h}*h*), so now it looks as if any reasonable person who hangs around with a human-like

*M*will soon know something that the machine itself can’t prove.

_{h}In my note, I argued that if *M _{h}* has an output very much like a person’s, then it must inevitably be that the number

*h*would be very hard for a human to describe. I suggested that the number

*h*would in fact be humanly unnameable in the sense that no person could in their lifetime produce an accurate description of this number.

My reason for wanting this to be the case was that, if *h* is humanly unnameable, then it would in fact be impossible for a person to precisely formulate, let alone assert, the sentences Tr(*h*) or Con(*h*). So the Lucas argument would fail to go through.

But now, having spent the last eighteen years teaching and researching computer science, I realize that we could in principle describe a rather simple “mind recipe” that could be used to evolve a computer program very much like a person—although simulating the evolution on computing machinery might well take very many years. The key point is that this mind recipe itself could be expressed by an easily nameable number *h.*

The basic idea behind the mind recipe is to start with a large population of sample programs, and to repeatedly measure their fitness at understanding some fixed library of books and movies, each time replacing the less fit programs with mutations or combinations of the more fit programs. If properly designed, then over time this simulated evolution has a chance of producing human-like machines. Why shouldn’t it? We ourselves evolved from a very humble start.

The mind recipe is both deterministic and nameable provided that, when we make mutations or choose pairs of programs to combine, we use a simply defined deterministic pseudorandomizer for the source of the random bits needed to orchestrate the process of simulated evolution. We might use, for instance, the traditional C language *rand*() function which returns the current value of a variable *random_integer* and at the same time refreshes this value to 214,013 · *random_integer* + 2,531,011. Alternately, one could use a one-dimensional cellular automaton such as the rule 30 that Stephen Wolfram describes in his monumental *A New Kind of Science* (Wolfram Media, 2002).

The point of the mind recipe is that we specify a complex machine *M* with a name of the form, say, “Seed your randomizer with the number 1946, create an initial population of a million random programs, and evolve the population through a billion generations, using the mutation and crossover genetic operators, and gauging fitness according to the following specific tests: answer the enclosed quizzes on the following books and movies, do better than the other programs at the following list of games, score well in the programs’ mutual rankings of each other, etc.” And this name can be transparently converted into a nameable code number *h.*

In other words, I was wrong in saying that if a machine *M _{h}* behaves like a human, then

*h*must be unnameable. Note that this doesn’t mean that the behavior of the end product

*M*is simple.

_{h}*M*itself is going to be the product of a billion generations of simulated evolution. The fully evolved program might, for instance, take the form of a neural network involving 100 trillion carefully tweaked real number weights—a mass of data that no human could hope to cogently overview.

_{h}*M*just happens to have a simple definition.

_{h}So now what about Lucas and Penrose? The philosopher Hilary Putnam formulated what remains the best counterargument in his 1960 essay “Minds and Machines,” which has been reprinted in A. R. Anderson, *Minds and Machines* (Prentice-Hall, 1964, pp. 43–59). (For Lucas’s ripostes to such objections, see his genial if unconvincing essay, “A paper read to the Turing Conference at Brighton on April 6th, 1990,” available online at http://users.ox.ac.uk/~jrlucas/Godel/brighton.html.)

Putnam’s point is simple. Even if you have seen *M _{h}* behaving sensibly for a period of time, you still don’t have any firm basis for asserting either that

*M*will always say only true things about mathematics or that

_{h}*M*will never fall into an inconsistency. Now if you were to have a full understanding of how

_{h}*M*operates, then perhaps you could prove that

_{h}*M*is consistent. But, as I just mentioned, in the case where

_{h}*h*is the mind recipe, the operation of the eventual

*M*is incomprehensibly intricate, and we will never be in a position to legitimately claim to know the truth of the sentence Con(

_{h}*h*), which asserts that

*M*is consistent. This is, indeed, the content of Gödel’s Second Incompleteness Theorem. Rather than ruling out man-machine equivalence, the theorem places limits on what we can know about machines equivalent to ourselves.

_{h}And, really, this shouldn’t come as a surprise. You can share an office or a house with a person P for fifteen years, growing confident in the belief that *P* is consistent, and then one day *P* begins saying and doing things that are completely insane. You imagined that you knew Con(*P*) to be true, but this was never the case at all. The only solid reason for asserting Con(*P*) would have been a systematic proof, but, given that you and *P* were of equivalent sophistication, this kind of proof remained always beyond your powers. All along, the very fact that Con(*P*) wasn’t provable contained the possibility that it wasn’t true. Like it or not, that’s the zone we operate in when relating to other intelligent beings.

On the topic of human and machine consciousness, I want to make an additional remark about the core “I am” sensation that I discuss in the “Robot Consciousness” section of Chapter Four. Somewhat disingenuously, I presented the “I am” sensation as being a given that results from simple existence. But it’s more realistic to say that a person’s core consciousness is based on certain kinds of activities taking place in the brain—processes which, by the way, could also be modeled by a program or a machine. A fairly detailed model of core consciousness appears in Antonio Damasio, *The Feeling of What Happens* (Harcourt, 1999). Damasio views consciousness as arising in the context of the following sequence:

• *Immersion.* You are active in the world.

• *Seeing Objects.* You distinguish separate objects in the world, including your body.

• *Movie-in-the-Brain.* You have an ongoing mental model of the world. The movie-in-the-brain includes images of the world’s objects and an image of your body.

• *Proto-Self.* Your image of your body differs from an image of an object in that your image of your body includes images of your sensations and current mental contents. This rich image is the proto-self.

• *Feelings.* You automatically and continually enhance the movie-in-the-brain by adding in representations of the proto-self’s interactions with objects. These second-order representations are what we call feelings.

• *Core Consciousness.* The act of continually forming feelings is part of what we mean by consciousness. At any given time, core consciousness is based on your feelings about a small group of images. Core consciousness highlights those particular images, which accounts for your current focus of attention.

You can find much more about robot intelligence, the mind recipe, and Damasio’s core consciousness in my forthcoming book, *The Lifebox*; *the Seashell and the Soul*, slated to appear from Thunder’s Mouth Press in 2005.

* * *

Set theorists have stayed busy during the twenty-five years since I wrote *Infinity and the Mind.* Recently my attention was caught by a pair of expository articles by the mathematician W. Hugh Woodin, “The Continuum Hypothesis, Part I” and “The Continuum Hypothesis, Part II” (*Notices of the American Mathematical Society*, 48(6):567–576 and 48(7):681–690, respectively). I was surprised and pleased to see that Woodin and some like-minded set theorists believe that Cantor’s continuum problem may in fact be solvable, and that the answer is likely to match Kurt Gödel’s conjecture that the size *c* of the continuum is the transfinite cardinal ℵ_{2}. During my years of exile among computing machines for whom 4 billion is the largest possible integer, I’d begun to fear that Cantor’s question about higher infinites might be meaningless.

A few weeks ago, I took a day off to visit Hugh Woodin in his office at the University of California in Berkeley, and he did his best to explain his recent work to me.

Woodin’s analysis can be formulated in terms of sets of the form *H*(*κ*), meaning the set of all sets hereditarily of cardinality less than *κ.* A set *x* is in *H*(*κ*) if the size of *x* is less than *κ*, and if all the members of *x* are in *H*(*κ*) as well, that is, the members of *x* have size less than *k, their* members have size less than *κ*, and so on. If we let ℵ_{0} be the first infinite ordinal, then *H*(ℵ_{0}) is the set of hereditarily finite sets. If there were no infinite sets at all, then the universe of set theory would be *H*(ℵ_{0}).

The analysis gets interesting when we look at *H*(ℵ_{1}), where ℵ_{1} is the first uncountable ordinal. *H*(ℵ_{1}) contains the hereditarily countable sets, that is, sets such that their size is countable or finite, their members are countable or finite, the members of their members are countable or finite, and so on. If there were no uncountable sets at all, then the universe of set theory would be *H*(ℵ_{1}). *H*(ℵ_{1}) can also be thought of as the universe of second-order number theory, where we talk about integers and sets of integers, for any set in *H*(ℵ_{1}) can be rather easily coded as a set of integers.

The first part of Woodin’s story relates to a set-theoretic axiom called projective determinacy (PD for short). The notion of projective determinacy has been around for several decades; in some sense it says that there are enough sets of natural numbers to satisfy each of a large range of conditions that one can formulate in *H*(ℵ_{1}). Saying that there are so many different sets of numbers means that the universe of set theory is wide, or rich in possibilities.

The new development is that work with large cardinal axioms has convinced many set theorists that projective determinacy is true. In other words, just as Kurt Gödel had hoped, thinking about higher orders of infinity has brought new insights about the structure of the smaller levels of sets such as *H*(ℵ_{1}). What makes this especially satisfying is that adding projective determinacy to the usual axioms of set theory (called ZFC for short) resolves many questions about the little universe *H*(ℵ_{1}) of hereditarily countable sets—and traditional Cohen forcing techniques cannot be used to prove that sentences in this universe are independent of projective determinacy. In Woodin’s terms, this means that the theory ZFC + PD is “generically absolute.”

The second part of Woodin’s story is about how set theorists have now turned to the logically next larger universe to look at: *H*(ℵ_{2}), the set of all sets with cardinality hereditarily less than ℵ_{2}. Given that being of size less than ℵ_{2} means being of size less than or equal to ℵ_{1}, we can say that *H*(ℵ_{2}) is the set of all sets *x* such that *x* has size less than or equal to ℵ_{1}, the set *x*’s members have size less than or equal to ℵ_{1}, and so on.

In the old days, rather than looking at the *H*() levels, it seemed more useful to look at the partial universes *V*_{α}, which are obtained by iterating the power set operation, as I describe on page 197. And, indeed, *V _{ω}* is identical to

*H*(ℵ

_{0}), and

*ν*is essentially the same as

_{ω+1}*H*(ℵ

_{1}). But, depending on how wide the universe of sets is, the step to

*V*can turn out to be a rather big jump that overshoots

_{ω}_{+2}*H*(ℵ

_{2}).

*H*(ℵ_{2}) is of crucial interest because Cantor’s Continuum Hypothesis can be formulated as a sentence in *H*(ℵ_{2})—this is because all maps from ℵ_{1} into the continuum are members of *H*(ℵ_{2}). If there were a way to exhaustively list the real numbers as a sequence of length this list would be present in *H*(ℵ_{2}).

Woodin’s current quest is to emulate the success with projective determinacy and to get a deeper understanding of this small universe of sets *H*(ℵ_{2}), with an eye to, among other things, solving Cantor’s continuum problem. And at this point he shifts into hyperdrive. He mixes extremely high-powered analysis of the set-theoretic universe as a whole with classical and detailed analysis of sets of natural numbers to arrive at the conclusion that *c* = ℵ_{2}.

But this conclusion is almost incidental to Woodin’s larger goals. He and his colleagues feel that set theory is at a crossroads relating to a principle that he calls the Ω Conjecture, and which I can barely understand. This is twenty-first-century mathematics, well beyond the grasp of my twentieth-century Ph.D. in set theory. No matter. The message to take away is that the study of infinity is alive and well, armed with powerful new techniques that make it stranger and more fascinating than ever.

Rudy Rucker

*Los Gatos, California*

*June 22, 2004*

## PREFACE TO THE PAPERBACK EDITION

Writing a second preface is a happy occasion.

This summer I was back in Heidelberg, with my beloved wife Sylvia, for the first time since 1980. Here’s an excerpt from my journal:

Being in Heidelberg makes me terribly nostalgic. Fifteen years ago I was a young man here, full of ideas, writing *Infinity and the Mind, White Light*, and *Software*, not to mention most of the stories in *The 57th Franz Kavka.* And the children were so sweet and young: tiny nursery-school Isabel, first-grade Rudy, Jr., dynamic ten-year-old Georgia. Sylvia was learning German and struggling against being a trapped housewife with the kids coming home from school at noon every day; it wasn’t an easy time for her. But for me—well, it seems like paradise, at least in retrospect. I thought I had so much time to finish my thinking. The ideas I developed were, in many respects, my complete and finished thoughts, e.g., on robot consciousness via evolution, but at the time I thought I was still just roughing out a start. I didn’t realize it was a high-water mark, and that I would never again think so deeply about the philosophy of mathematics. The Liar Paradox, the Berry Paradox—I solved them all to my own satisfaction; I got them to stop itching at me. And I created a kind of para-solution to the Continuum Problem in the form of my novel *White Light.*

Thoughts today looking over *Infinity and the Mind:*

Cantor’s theory of higher infinities and his discovery of the Continuum Problem remain, at least for this mathematician, The Greatest Story Ever Told.

I am a satisfied user of the solution to the classic paradoxes presented in “The Unnameable.” Proof: I was able to stop thinking about the paradoxes. (See the table on p. 153 for a summary.)

The “Robots and Souls” discussion of human-machine equivalence is, I think, definitive. (See p. 171 for Gödel’s seed-quote, p. 183 for my position, and p. 292 for a formal refutation of the Lucas-Penrose argument.)

Chapter Five’s contents have been field-tested. Conclusion: All is One. Thank you for Everything, dear Absolute. Praise all in the channel, and let all praise the One!

Rudy Rucker

*Los Gatos, California*

*January 11, 1995*

## PREFACE

This book discusses every kind of infinity: potential and actual, mathematical and physical, theological and mundane. Talking about infinity leads to many fascinating paradoxes. By closely examining these paradoxes we learn a great deal about the human mind, its powers, and its limitations.

The study of infinity is much more than a dry, academic game. The intellectual pursuit of the Absolute Infinite is, as Georg Cantor realized, a form of the soul’s quest for God. Whether or not the goal is ever reached, an awareness of the process brings enlightenment.

*Infinity and the Mind* has been written with the average person in mind. Most of the main text should prove digestible, if chewed. By and large, the separate sections are complete in themselves, and the reader should feel free to skip about in the book.

At the end of each chapter there is a section with puzzles and paradoxes; answers are provided. For those who may wish to delve a bit deeper into set theory and logic, I have organized two mathematical excursions which are placed at the end of the book.

*Infinity and the Mind* was thought out and written over a period of some ten years. I started having ideas for it in that most Sixties of years, 1972. At that time I was writing a doctoral dissertation in set theory for Erik Ellentuck at Rutgers University and attending a logic seminar led by the eminent proof theorist Gaisi Takeuti at the Institute for Advanced Study in Princeton, New Jersey. The first time I met Takeuti I asked him what set theory was really about. “We are trying to get exact description of thoughts of infinite mind,” he said. And then he laughed, as if filled with happiness by this impossible task.

The same year I met Kurt Gödel at the Institute for Advanced Study. No one in modern times has thought more logically than Gödel, no one has proved theorems of greater mathematical complexity. Yet the man I met was a joyful, twinkling sage-not some obsessed fossil. What struck me most about Gödel was his intellectual freedom-his ability to move back and forth between frankly mystical insights and utterly precise logical derivations. As I began to study the writings of Georg Cantor, the founder of set theory, I realized that Cantor shared this freedom. Logic and set theory are the tools for an exact metaphysics.

The writing of this book started with a paper I did for a logic colloquium at Oxford University in 1976 and began in earnest with a set of mimeographed lecture notes for an interdepartmental course I taught with my friend William J. Edgar at SUNY Geneseo in 1977. In 1978 I rewrote my notes and reproduced them by photo-offset for an experimental metamathematics course. Those notes make up the present Chapters One and Three and the more technical Excursion I.

I spent the years 1978-1980 at the Mathematics Institute of the University of Heidelberg, a guest of Gert Müller and the Alexander von Humboldt Foundation. While there, I wrote Chapter Four with Excursion II for a course of lectures on the philosophy of mathematics. Chapters Two and Five have been written this winter at Randolph-Macon Woman’s College.

*Infinity and the Mind* is a work of transmission. I dedicate it with love and respect to everyone in the channel.

R.v.B.R.

*Lynchburg, Virginia*

*June 19, 1981*

# INFINITY

# and the

# MIND

## CHAPTER ONE

## INFINITY

### A SHORT HISTORY OF INFINITY

*This is a free online webpage edition. Copyright (C) Rudy Rucker 2016.
Purchase paperback and ebook editions from*

The symbol for infinity that one sees most often is the lazy eight curve, technically called the lemniscate. This symbol was first used in a seventeenth century treatise on conic sections.1 It caught on quickly and was soon used to symbolize infinity or eternity in a variety of contexts. For instance, in the 1700s the infinity symbol began appearing on the Tarot card known as the Juggler or the Magus. It is an interesting coincidence that the Qabbalistic symbol associated with this particular Tarot card is the Hebrew letter ℵ, (pronounced alef), for Georg Cantor, the founder of the modern mathematical theory of the infinite, used the symbol ℵ_{0}, (pronounced alef-null), to stand for the first infinite number.

The appropriateness of the symbol ∞ for infinity lies in the fact that one can travel endlessly around such a curve … demolition derby style, if you will. Endlessness is, after all, a principal component of one’s concept of infinity. Other notions associated with infinity are indefiniteness and inconceivability.

Infinity commonly inspires feelings of awe, futility, and fear. Who as a child did not lie in bed filled with a slowly mounting terror while sinking into the idea of a universe that goes on and on, for ever and ever? Blaise Pascal puts this feeling very well: “When I consider the small span of my life absorbed in the eternity of all time, or the small part of space which I can touch or see engulfed by the infinite immensity of spaces that I know not and that know me not, I am frightened and astonished to see myself here instead of there … now instead of then.”2

It is possible to regard the history of the foundations of mathematics as a progressive enlarging of the mathematical universe to include more and more infinities. The Greek word for infinity was *apeiron*, which literally means unbounded, but can also mean infinite, indefinite, or undefined. *Apeiron* was a negative, even pejorative, word. The original chaos out of which the world was formed was *apeiron.* An arbitrary crooked line was *apeiron*. A dirty crumpled handkerchief was *apeiron*. Thus, *apeiron* need not only mean infinitely large, but can also mean totally disordered, infinitely complex, subject to no finite determination. In Aristotle’s words, “… being infinite is a privation, not a perfection but the absence of a limit….”3

There was no place for the *apeiron* in the universe of Pythagoras and Plato. Pythagoras believed that any given aspect of the world could be represented by a finite arrangement of natural numbers, (where “natural number” means “whole number.”) Plato believed that even his ultimate form, the Good, must be finite and definite. This was in contradistinction to almost all later metaphysicians, who assumed that the Absolute is necessarily infinite. In the next chapter I will discuss the way in which Greek mathematics was limited by this refusal to accept the *apeiron*, even in the relatively harmless guise of a real number with an infinite decimal expansion.

Aristotle recognized that there are many aspects of the world that seem to point to the actuality of the *apeiron.* For instance, it seems possible that time will go on forever; and it would seem that space is infinitely divisible, so that any line segment contains an infinity of points. In order to avoid these actual infinites that seemed to threaten the orderliness of his *a priori* finite world, Aristotle invented the notion of the *potentially infinite* as opposed to the *actually infinite.* I will describe this distinction in more detail in the next section, but for now let me characterize it as follows. Aristotle would say that the set of natural numbers is potentially infinite, since there is no largest natural number, but he would deny that the set is actually infinite, since it does not exist as one finished thing. This is a doubtful distinction, and I am inclined to agree with Cantor’s opinion that “… in truth the potentially infinite has only a borrowed reality, insofar as a potentially infinite concept always points towards a logically prior actually infinite concept whose existence it depends on.”4

Plotinus was the first thinker after Plato to adopt the belief that at least God, or the One, is infinite, stating of the One that, “Absolutely One, it has never known measure and stands outside of number, and so is under no limit either in regard to anything external or internal; for any such determination would bring something of the dual into it.”5

St. Augustine, who adapted the Platonic philosophy to the Christian religion, believed not only that God was infinite, but also that God could think infinite thoughts. St. Augustine argued that, “Such as say that things infinite are past God’s knowledge may just as well leap head long into this pit of impiety, and say that God knows not all numbers…. What madman would say so? … What are we mean wretches that dare presume to limit His knowledge?”6

This extremely modern position will be returned to in the last section of this chapter. Later medieval thinkers did not go as far as Augustine and, although granting the unlimitedness of God, were unwilling to grant that any of God’s creatures could be infinite. In his *Summa Theologiae* St. Thomas Aquinas gives a sort of Aristotelian proof that “although God’s power is unlimited, he still cannot make an absolutely unlimited thing, no more than he can make an unmade thing (for this involves contradictories being true together).”7 The arguments are elegant, but suffer from the flaw of being circular: it is proved that the notion of an unlimited thing is contradictory by slipping in the premise that a “thing” is by its very nature limited.

Thus, with the exception of Augustine and a few others, the medieval thinkers were not prepared to deal with the infinitude of any entities other than God, be they physical, psychological, or purely abstract. The famous puzzle of how many angels can dance on the head of a pin can be viewed as a question about the relationship between the infinite Creator and the finite world. The crux of this problem is that, on the one hand, it would seem that since God is infinitely powerful, he should be able to bid an infinite number of angels to dance on the head of a pin; on the other hand, it was believed by the medieval thinkers that no actually infinite collection could ever arise in the created world.

Their proofs that infinity is somehow a self-contradictory notion were all flawed, but there was at least one interesting paradox involving infinity that the medieval thinkers were aware of. It would seem that any line includes infinitely many points. Since the circumference of a circle with radius two is two times as long as the circumference of a circle with radius one, then the former should include a *larger* infinity of points than the latter. But by drawing radii we can see that each point *P* on the small circle corresponds to exactly one point *P*′ on the large circle, and each point *Q*′ on the large circle corresponds to exactly one point *Q* on the small circle. Thus we seem to have two infinities that are simultaneously different and equal.

In the early 1600s Galileo Galilei offered a curious solution to this problem. Galileo proposed that the smaller length could be turned into the longer length by adding an infinite number of infinitely small gaps. He was well aware that such a procedure leads to various difficulties: “These difficulties are real; and they are not the only ones. But let us remember that we are dealing with infinites and indivisibles, both of which transcend our finite understanding, the former on account of their magnitude, the latter because of their smallness. In spite of this, men cannot refrain from discussing them, even though it must be done in a roundabout way.”8

He resolved some of his difficulties by asserting that problems arise only, “when we attempt, with our finite minds, to discuss the infinite, assigning to it those properties which we give to the finite and limited; but this I think is wrong, for we cannot speak of infinite quantities as being the one greater or less than or equal to another.”9 This last assertion is supported by an example that is sometimes called Galileo’s paradox.

The paradoxical situation arises because, on the one hand, it seems evident that most natural numbers are not perfect squares, so that the set of perfect squares is smaller than the set of all natural numbers; but, on the other hand, since every natural number is the square root of exactly one perfect square, it would seem that there are just as many perfect squares as natural numbers. For Galileo the upshot of this paradox was that, “we can only infer that the totality of all numbers is infinite, and that the number of squares is infinite …; neither is the number of squares less than the totality of all numbers, nor the latter greater than the former; and finally, the attributes ‘equal,’ ‘greater,’ and ‘less,’ are not applicable to infinite, but only to finite quantities.”10

I have quoted Galileo at some length, because it is with him that we have the first signs of the modern attitude toward the actual infinite in mathematics. If infinite sets do not behave like finite sets, this does not mean that infinity is an inconsistent notion. It means, rather, that infinite numbers obey a different “arithmetic” from finite numbers. If using the ordinary notions of “equal” and “less than” on infinite sets leads to contradictions, this is not a sign that infinite sets cannot exist, but, rather, that these notions do not apply without modification to infinite sets. Galileo himself did not see how to carry out such a modification of these notions; *this* was to be the task of Georg Cantor, some 250 years later.

One of the reasons that Galileo felt it necessary to come to some sort of terms with the actual infinite was his desire to treat space and time as continuously varying quantities. Thus, the results of an experiment on motion can be stated in the form that *x* = *f*(*t*), that space position is a certain function of continuously changing time. But this variable *t* that grows continuously from, say, zero to ten is *apeiron*, both in the sense that it takes on arbitrary values, and in the sense that it takes on infinitely many values.

This view of position as a function of time introduced a problem that helped lead to the founding of the Calculus in the late 1600s. The problem was that of finding the instantaneous velocity of a moving body, whose distance *x* from its starting point is given as a function *f*(*t*) of time.

It turns out that to calculate the velocity at some instant *t*_{0}, one has to imagine measuring the speed over an infinitely small time interval *dt.* The speed *f*′(*t*_{0}) at *t*_{0} is given by the formula (*f*(*t*_{0} + *dt*) − *f*(*t*_{0}))/*dt*, as everyone who has ever survived a first-year calculus course knows.

The quantity *dt* is called an *infinitesimal*, and obeys many strange rules. If *dt* is added to a regular number, then it can be ignored, treated like zero. But, on the other hand, *dt* is regarded as being different enough from zero to be usable as the denominator of a fraction. So is *dt* zero or not? Adding finitely many infinitesimals together just gives another infinitesimal. But adding infinitely many of them together can give either an ordinary number, or an infinitely large quantity.

Bishop Berkeley found it curious that mathematicians could swallow the Newton-Leibniz theory of infinitesimals, yet balk at the peculiarities of orthodox Christian doctrine. He wrote about this in a 1734 work, the full title of which was, *The Analyst, Or A Discourse Addressed to an Infidel Mathematician. Wherein It is examined whether the Object, Principles, and Inferences of the modern Analysis are more distinctly conceived, or more evidently deduced, than Religious Mysteries and Points of Faith. “First cast out the beam out of thine own Eye; and then shalt thou see clearly to cast out the mote out of thy brother’s Eye.”11*

The use of infinitely small and infinitely large numbers in calculus was soon replaced by the limit process. But it is unlikely that the Calculus could ever have developed so rapidly if mathematicians had not been willing to think in terms of actual infinities. In the past fifteen years, Abraham Robinson’s non-standard analysis has produced a technique by which infinitesimals can be used without fear of contradiction. Robinson’s technique involves enlarging the real numbers to the set of *hyperreal* numbers, which will be discussed in Chapter 2.

After the introduction of the limit process, calculus was able to advance for a long time without the use of any actually infinite quantities. But as mathematicians tried to get a precise *description* of the continuum or real line, it became evident that infinities in the foundations of mathematics could only be avoided at the cost of great artificiality. Mathematicians, however, still hesitated to plunge into the world of the actually infinite, where a set could be the same size as a subset, a line could have as many points as a line half as long, and endless processes were treated as finished things.

In was Georg Cantor who, in the late 1800s, finally created a theory of the actual infinite which by its apparent consistency, demolished the Aristotelian and scholastic “proofs” that no such theory could be found. Although Cantor was a thoroughgoing scholar who later wrote some very interesting philosophical defenses of the actual infinite, his point of entry was a mathematical problem having to do with the uniqueness of the representation of a function as a trigonometric series.

To give the flavor of the type of construction Cantor was working with, let us consider the construction of the Koch curve shown in Figure 4. The Koch curve is found as the limit of an infinite sequence of approximations. The first approximation is a straight line segment (stage 0). The middle third of this segment is then replaced by two pieces, each as long as the middle third, which are joined like two sides of an equilateral triangle (stage 1). At each succeeding stage, each line segment has its middle third replaced by a spike resembling an equilateral triangle.

Now, if we take infinity as something that can, in some sense, be attained, then we will regard the limit of this infinite process as being a curve actually existing, if not in physical space, then at least as a mathematical object. The Koch curve is discussed at length in Benoit Mandelbrot’s book, *Fractals*, where he explains why there is reason to think of the Koch curve in its infinite spikiness as being a better model of a coastline than any of its finitely spiky approximations.12

Cantor soon obtained a number of interesting results about actually infinite sets, most notably the result that the set of points on the real line constitutes a *higher* infinity than the set of all natural numbers. That is, Cantor was able to show that infinity is not an all or nothing concept: there are degrees of infinity.

This fact runs counter to the naive concept of infinity: there is only one infinity, and this infinity is unattainable and not quite real. Cantor keeps this naive infinity, which he calls the Absolute Infinite, but he allows for many intermediate levels between the finite and the Absolute Infinite. These intermediate stages correspond to his *transfinite numbers …* numbers that are infinite, but none the less conceivable.

In the next section we will discuss the possibility of finding *physically* existing transfinite sets. We will then look for ways in which such actual infinities might exist *mentally.* Finally we will discuss the Absolute, or metaphysical, infinite.

This threefold division is due to Cantor, who, in the following passage, distinguishes between the Absolute Infinite, the physical infinities, and the mathematical infinities:

The actual infinite arises in three contexts: *first* when it is realized in the most complete form, in a fully independent other-worldly being, *in Deo*, where I call it the Absolute Infinite or simply Absolute; *second* when it occurs in the contingent, created world; *third* when the mind grasps it *in abstracto* as a mathematical magnitude, number, or order type. I wish to make a sharp contrast between the Absolute and what I call the Transfinite, that is, the actual infinities of the last two sorts, which are clearly limited, subject to further increase, and thus related to the finite.13

### PHYSICAL INFINITIES14

*This is a free online webpage edition. Copyright (C) Rudy Rucker 2016.
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There are three ways in which our world appears to be unbounded and thus, perhaps, infinite. It seems that time cannot end. It seems that space cannot end. And it seems that any interval of space or time can be divided and subdivided endlessly. We will consider these three apparent physical infinites in three subsections.

#### TEMPORAL INFINITIES

Suppose that the human race was never going to die out—that any given generation would be followed by another generation. Would we not then have to admit that the number of generations of man is actually infinite?

Aristotle argued against this conclusion, asserting that in this situation the number of generations of man would be but potentially infinite; that is, infinite only in the sense of being inexhaustible. He maintained that at any given time there would only have been some finite number of generations, and that it was not permissible to take the entire future as a single whole containing an actual infinitude of generations.

It is my opinion that this sort of distinction rests on a view of time that has been fairly well discredited by modern relativistic physics. In order to agree with Aristotle that, although there will never be a last generation, there is no infinite set of all the generations, we must believe that the future does not exist as a stable, definite thing. For if we have the future existing in a fixed way, then we have all of the infinitely many future generations existing “at once.”

But one of the chief consequences of Einstein’s Special Theory of Relativity is that it is space-time that is fundamental, not isolated space which evolves as time passes. I will not argue this point in detail here, but let me repeat that on the basis of modern physical theory we have every reason to think of the passage of time as an illusion. Past, present, and future all exist together in space-time.

So the question of the infinitude of time is not one that is to be dodged by denying that time can be treated as a fixed dimension such as space. The question still remains: is time infinite? If we take the entire space-time of our universe, is the time dimension infinitely extended or not?

Fifty, or even twenty, years ago it would have been natural to assert that our universe has no beginning or end and that time is thus infinite in both directions. But recently it has become an established fact that the universe does have a beginning in time known as the Big Bang. The Big Bang took place approximately 15 billion years ago. At that time our universe was the size of a point, and it has been expanding ever since. What happened before the Big Bang? It is at least possible to answer, “Nothing.” The apparent paradox of having a *first* instant in time is sometimes avoided by saying that the Big Bang did not occur *in* time … that time is open, rather than closed, in the past.

This is a subtle distinction, but a useful one. If we think of time as being all the points greater than or equal to zero, then there is a first instant: zero. But if we think of time as being all the points strictly greater than zero, then there is no first instant. For any instant *t* greater than zero, one has an earlier instant *t/*2 that is also greater than zero.

But in any case, if we think of time as not existing before the Big Bang, then there are certainly not an infinite number of years in our past. And what about the future? There is no real consensus on this. Many cosmologists feel that our universe will eventually stop expanding and collapse to form a single huge black hole called the Big Stop or the Gnab Gib; others feel that the expansion of the universe will continue indefinitely.

If the universe really does start as a point and eventually contract back to a point, is it really reasonable to say that there is no time except for the interval between these points? What comes before the beginning and after the end?

One response is to view the universe as an oscillating system, which repeatedly goes through expansions and contractions. This would reintroduce an infinite time, which could, however, be avoided.

The way in which one would avoid infinite time in an endlessly oscillating universe would be to adopt a belief in what used to be called “the eternal return.” This is the belief that every so often the universe must repeat itself. The idea is that a finite universe must return to the same state every so often, and that once the same state has arisen, the future evolution of the universe will be the same as the one already undergone. The doctrine of eternal recurrence amounts to the assumption that time is a vast circle. An oscillating universe with circular time is pictured in Figure 10.

There is a simpler model of an oscillating universe with circular time, which can be called *toroidal space-time.* In toroidal space-time we have an oscillating universe that repeats itself after every cycle. Such a model is obtainable by identifying the two points, “Big Bang” and “Big Stop,” in Figure 11.

Note, however, that if the universe really *expands* forever, then it cannot ever repeat itself, as the average distance between galaxies is a continually increasing quantity that never returns to the same value.

#### SPATIAL INFINITIES

We now turn to a consideration of the possibility of spatial infinites. The potential versus actual infinity distinction is sometimes used to try to scotch this question at the outset. Immanuel Kant, for instance, argues that the world cannot be an infinite whole of coexisting things because “in order therefore to conceive the world, which fills all space, as a whole, the successive synthesis of the parts of an infinite world would have to be looked upon as completed; that is, an infinite time would have to be looked upon as elapsed, during the enumeration of all coexisting things.”15

Kant’s point is that space is in some sense not already really there—that things exist together in space only when a mind perceives them to do so. If we accept this, then it is true that an infinite space is something that no finite mind can know of after any finite amount of time. But one feels that the world does exist as a whole, in advance of any efforts on our part to see it as a unity. And if we take all of space-time, it certainly does not seem to be meaningless to ask whether the spatial extent of space-time is infinite or not.

In *De Rerum Natura*, Lucretius first gave the classic argument for the unboundedness of space: “Suppose for a moment that the whole of space were bounded and that someone made his way to its uttermost boundary and threw a flying dart.”16 It seems that either the dart must go past the boundary, in which case it is no boundary of space; or the dart must stop, in which case there is something just beyond the boundary that stops it, which again means that the purported boundary is not really the end of the universe.

So great was their revulsion against the *apeiron* that Parmenides, Plato, and Aristotle all held that the space of our universe is bounded and finite, having the form of a vast sphere. When faced with the question of what lies outside this sphere, Aristotle maintained that “what is limited, is not limited in reference to something that surrounds it.”17

In modern times we have actually developed a way to make Aristotle’s claim a bit more reasonable. As Lucretius realized, the weak point in the claim that space is a finite sphere is that such a space has a definite boundary. But there is a way to construct a three-dimensional space which is finite and which does *not* have boundary points: simply take the hypersurface of a hypersphere. Such a space is endless but not infinite.

To understand how something can be endless but not infinite, think of a circle. A fly can walk around and around the rim of a glass without ever coming to a barrier or stopping point, but none the less he will soon retrace his steps.

Again, the surface of the Earth is a two-dimensional manifold which is finite but unbounded (unbounded in the sense of having no edges). You can travel and travel on the Earth’s surface without ever coming to any truly impassible barrier … but if you continue long enough, you will begin to recross your steps.

The reason that the two-dimensional surface of the Earth is finite but unbounded is that it is bent, in three-dimensional space, into the shape of a sphere. In the same way, it is possible to imagine the three-dimensional space of our universe as being bent, in some four-dimensional space, into the shape of a hypersphere. It was Bernhard Riemann who first realized this possibility in 1854. There is, however, a traditional belief that anticipates the hypersphere. This tradition, described in the essay, “The Fearful Sphere of Pascal,” by Jorge Luis Borges, is summarized by the saying (attributed to the legendary magician Hermes Trismegistus) that “God is an intelligible sphere, whose center is everywhere and whose circumference is nowhere.”18 If the universe is indeed a hypersphere, then it would be quite accurate to regard it as a sphere whose center is everywhere and whose circumference is nowhere.

To see why this is so, consider the fact that if space is hyperspherical, then one can cover all of space by starting at any point and letting a sphere expand outwards from that point. The curious thing is that if one lets a sphere expand in hyperspherical space, there comes a time when the circumference of the sphere turns into a point and disappears. This fact can be grasped by considering the analogous situation of the sequence of circular latitude lines on the spherical surface of the earth.19 This line of thought appears in Dante’s *Paradisio* (1300).20

Aristotle had believed that the world was a series of nine spheres centered around the Earth. The last of these crystalline spheres was called the *Primum Mobile* and lay beyond the sphere upon which were fastened all of the stars (other than the sun, which was attached to the fourth sphere). In the *Paradisio*, Dante is led out through space by Beatrice. He passes through each of the nine spheres of the world: Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn, Fixed Stars, Primum Mobile. Beyond these nine spheres lie nine spheres of angels, corresponding to the nine spheres of the world. Beyond the nine spheres of angels lies a *point* called the Empyrean, which is the abode of God.

The puzzling thing about Dante’s cosmos as it is drawn in Figure 14 is that here the Empyrean appears not to be a point, but rather to be all of space (except for the interior of the last sphere of angels). But this can be remedied if we take space to be hyperspherical! In Figure 15 I have drawn the model we obtain if we take the diagram on the last page and curve it up into a sphere with a point-sized Empyrean. In the same way, the three-dimensional model depicted by the first picture can be turned into the finite unbounded space of the second picture if we bend our three-dimensional space in such a way that all of the space outside our last angelic sphere is compressed to a point.21 Figure 16 is Doré’s engraving of the Empyrean surrounded by its spheres of angels.

This whole notion of hyperspherical space was not consciously developed until the mid-nineteenth century. In the Middle Ages there was a general and uncritical acceptance of Aristotle’s view of the universe-without Dante’s angelic spheres.

Lucretius, of course, had insisted that space is infinite, and there were many other thinkers, such as Nicolas of Cusa and Giordano Bruno, who believed in the infinitude of space. Some kept to the Aristotelian world system, but suggested that there were many such setups drifting around; others opted for a looser setup under which stars and planets are more or less randomly mixed together in infinite space.

Bruno strongly advocated such viewpoints in his writings, especially his dialogue of 1584, “On the Infinite Universe and Worlds.”22 Bruno travelled freely around Europe during his lifetime, teaching his doctrine of the infinite universe at many centers of learning. In 1591, a wealthy Venetian persuaded Bruno to come from Frankfurt to teach him “the art of memory and invention.” Shortly after Bruno arrived, the trap was sprung. His host had been working closely with the ecclesiastical authorities, who considered Bruno a leading heretic or heresiarch. Bruno was turned over to the Inquisition. For nine years Bruno was interrogated, tortured, and tried, but he would not give up his beliefs; early in 1600 he was burned at the stake in the Roman *Piazza* Campo di Fiori. Bruno’s example caused Galileo to express himself a good deal more cautiously on scientific questions in which the Church had an interest.

Whether or not our space is actually infinite is a question that could conceivably be resolved in the next few decades. Assuming that Einstein’s theory of gravitation is correct, there are basically two types of universe: i) a hyperspherical (closed and unbounded) space that expands and then contracts back to a point; ii) an infinite space that expands forever. It is my guess that case i) will come to be most widely accepted, if only because the notion of an actually infinite space extending out in every direction is so unsettling. The fate of the universe in case i) is certainly more interesting, since such a universe collapses back to an infinitely dense space-time singularity that may serve as the seed for a whole new universe. In case ii), on the other hand, we simply have cooling and dying suns drifting further and further apart in an utterly empty black immensity … and in the end there are only ashes and cinders in an absolute and eternal night.

Even though I am basically pro-infinity, my emotions lie with the hyperspherical space. But is there any way of finding a spatial infinity here? Well, what about that four-dimensional space in which our hyperspherical universe is floating? Many would dismiss this space as a mere mathematical fiction … as a colorful way of expressing the finite, but unbounded, nature of our universe. This widely held position is really a more sophisticated version of Aristotle’s claim that what is limited need not be limited with reference to something outside itself.

But what if one chooses to believe that the four-dimensional space in which our universe curves is real? We might imagine a higher 4-D (four-dimensional) world called, let us say, a duoverse. The duoverse would be 4-D space in which a number of hyperspheres were floating. The hypersurface of each of the hyperspheres would be a finite, unbounded 3-D universe.

Thus, a duoverse would contain a number of 3-D universes, but no inhabitant of any one of these universes could reach any one of the others, unless he could somehow travel through 4-D space. By lowering all the dimensions by one, one can see that this situation is analogous to a universe that is a 3-D space in which a number of spheres are floating. The surface of each sphere or planet is a finite, unbounded 2-D space; and no one can get from one planet surface to another planet surface without travelling through 3-D space.

Following the Hermetic principle, “As above, so below,” one is tempted to believe that the duoverse we are in is actually a finite and unbounded 4-D space (the 4-D surface of a 5-D sphere in 5-D space), and that there are a number of such duoverses drifting about in a 5-D triverse. This could be continued indefinitely. One is reminded of those Eastern descriptions of the world as a disk resting on the backs of elephants, who stand upon a turtle, who stands upon a turtle, who stands upon a turtle, who stands upon a turtle, etc.

Note that in that particular sort of cosmos there is only one universe, one duoverse, one triverse, and so on. But in the kind of infinitely regressing cosmos that I have drawn in Figure 18, we have infinitely many objects at each level. Note also that to get from star A to star B one would have to move through 5-D space to get to a different duoverse. It is a curious feature of such a cosmos that, although there are an infinite number of stars, no one *n*-dimensional space has more than a finite number of them.

The question we are concerned with here is whether or not space is infinitely large. There seem to be three options: i) There is some level *n* for which *n*-dimensional space is real and infinitely extended. The situation where our three-dimensional space is infinitely large falls under this case. ii) There is some *n* such that there is only one *n*-dimensional space. This space is to be finite and unbounded, and there is to be no reality to *n* + 1 dimensional space. The situation where our three-dimensional space is finite and unbounded, and the reality of four-dimensional space denied, falls under this case. iii) There are real spaces of every dimension, and each of these spaces is finite and unbounded. In this case we either have an infinite number of universes, duoverses, etc., or we reach a level after which there is only one *n*-verse for each *n.*

So is space infinite? It seems that we can insist that at some dimensional level it is infinite; adopt the Aristotelian stance that space is finite at some level beyond which nothing lies; or accept the view that there is an infinite sequence of dimensional levels. In this last case we already have a qualitative infinity in the dimensionality of space, and we may or may not have a quantitative infinity in terms, say, of the total volume of all the 3-D spaces involved.

#### INFINITIES IN THE SMALL

In this subsection I will discuss the existence of the infinity in the *small*, as opposed to the infinity in the *large*, which has just been discussed. Since a point has no length, no finite number of points could ever constitute a line segment, which *does* have length. So it seems evident that every line segment, or, for that matter, every continuous plane segment or region of space, must consist of an infinite number of points. By the same token, any interval of time should consist of an infinite number of instants; and any continuous region of space-time would consist of an infinite number of events (*event* being the technical term for a space-time location, i.e., point at an instant).

It is undeniable that a continuous region of *mathematical* space has an infinite number of *mathematical* points. Right now, however, we are concerned with *physical* space. We should not be too hasty in assuming that every property of the abstract mathematical space we use to organize our experiences is an actual property of the concrete physical space we live in. But what *is* “the space we live in”? If it is not the space of mathematical physics, is it the space of material objects? Is it the space of our perceptions? In terms of material objects or of perceptions, points do not really exist; for any material or perceptual phenomenon is spread over a certain finite region of space-time. So when we look for the infinity in the small in matter, we do not ask whether matter consists of an infinity of (unobservable) mass-points, but, rather, whether matter is infinitely divisible.

A commitment to avoiding the formless made it natural for Greek atomists such as Democritus to adopt a theory of matter under which the seemingly irregular bodies of the world are in fact collections of indivisible, perfectly formed atoms. (The four kinds of atoms were shaped, according to Plato, like four of the regular polyhedra. There is one other polyhedron, the twelve-sided dodecahedron, and this was thought somehow to represent the Universe with its twelve signs of the zodiac.) For the atomists, it was as if the world were an immense Lego set, with four kinds of blocks. The diverse substances of the world-oil, wood, stone, metal, flesh, wine, and so on-were regarded as being mixtures of the four elemental substances: Earth, Air, Fire, and Water. Thus, gold was regarded by Plato as being a very dense sort of Water, and copper was viewed as gold with a small amount of Earth mixed in.

The alchemists and early chemists adopted a similar system, only the number of elemental substances became vastly enlarged to include all homogeneous substances, such as the various ores, salts, and essences. The fundamental unit here was the molecule.

A new stage in man’s conception of matter came when it was discovered that if an electric current is passed through water, it can be decomposed into hydrogen and oxygen. Eventually, the vast diversity of existing molecules was brought under control by regarding molecules as collections of atoms. Soon some ninety different types of atoms or chemical elements were known. A new simplification occurred when it was discovered, by bombarding a sheet of foil with alpha rays, that an atom consists of a positive nucleus surrounded by electrons. Shortly after this the neutron was discovered, and the physical properties of the various atoms were accounted for by regarding them as collections of protons, neutrons, and electrons.

Over the last half century it has been learned, by using particle accelerators, that there are actually many types of “elementary particles” other than the neutron, electron, and proton. The situation in high-energy physics today is as follows. A few particles–electrons, neutrinos, and muons–seem to be absolutely indivisible. These particles are called *leptons.* All others–protons, neutrons, mesons, lambdas, etc.–can be broken up into smaller units, which then reassemble to form more particles.

The historical pattern in the investigation of matter has been the explanation of diverse substances as combinations of a few simpler substances. Diversity of form replaces diversity of substance. So it is no surprise that it has been proposed that the great variety of divisible particles that exist can be accounted for by assuming that these particles are all built up out of *quarks.*

A second element in the historical pattern is that as more powerful tools of investigation are used, it becomes evident that there are more types of new building blocks than had been suspected initially. This is the phase that high-energy physics is currently moving into. First there were three kinds of quark: up, down, and strange. Now, the charmed quark has been admitted, and there are two new possible quarks: the top quark and the bottom quark. It seems likely that the many diverse types of quark will eventually be accounted for by assuming that each quark is a combination of a few, let us say, *darks …* and that there are only a very small number of possible kinds of dark. The cycle will then repeat, with more and more different sorts of dark being indirectly observed, the new diversity being accounted for by viewing each dark as a collection of a few smaller particles of which there are a limited variety, this limited variety beginning to proliferate, and so on.

If this sort of development can indeed continue indefinitely, then we are left with the fact that a stone is a collection of collections of collections of…. The stone thus consists of an infinite number of particles, no one of which is indivisible. There is, finally, no matter—only form. For a stone is mostly empty space with a few molecules in it, a molecule is a cloud of atoms, an atom is a few electrons circling a tiny nucleus…. What if any seemingly solid bit of matter proves on closer inspection to be a cloud of smaller bits of matter, which are in turn clouds, and so on? Note that the branching matter tree that I began to draw for the stone has only a finite number of forks or nodes at each level, but that since there are infinitely many levels, there are in all an infinite number of nodes or component particles.

There are various objections to this sort of physical infinity. One is the Aristotelian argument that unless one is actually smashing the stone down to the quark level, the quarks are only potentially (as opposed to actually) there. The point would be that the stone may be indefinitely divisible, but that since no one will ever carry out infinitely many divisions, there are not really infinite numbers of particles in the stone right now.

There is a more practical objection as well. This is that no quark has ever been observed in isolation; the existence of quarks is deduced only indirectly as a way of explaining the symmetries of structure that occur in tables of the elementary particles. This argument is not very strong, however. For one thing, a great number of the things we believe in can be observed only indirectly; and, more practically, if we can continue to increase the energy of our measuring tools, there is no reason to think that quarks cannot be more convincingly detected.

A more fundamental objection to the whole idea of particles, subparticles, etc., is that the underlying reality of the world may be field-like, rather than particle-like. By splitting particles indefinitely we arrived at the conclusion that there is only form, and no content; many physicists prefer to start with this viewpoint. For these physicists, the various features of the world are to be explained in terms of the geometry of space-time. To get a feeling for this viewpoint, one should look carefully at the surface of a river or small brook. There are circular ripples, flow bulges, whirlpools and eddies, bubbles that form, drops that fly up and fall back, waves that crest into foam. The *geometrodynamic* world-view regards space-time as a substance like the surface of a brook; the various fields and particles that seem to exist are explained as features of the flow.

Does the space-time of geometrodynamics allow an infinity in the small? There is really no answer to this question at present. According to one viewpoint there should be a sort of graininess to space-time, and the grain size would represent a sort of indivisible atom; a different viewpoint suggests that space-time should be as infinitely continuous as mathematical space.

What if there really is nothing smaller than electrons and quarks? Is there then any hope of an infinity in the small? One can argue that a given electron can have infinitely many locations along a given meter stick, so that our space really does have infinitely many points. It is sometimes asserted that the uncertainty principle of quantum mechanics nullifies this argument, but this is not the case.

Quantum mechanics puts no upper limit on the precision with which one can, in principle, determine the position of an electron. It is just that the more precisely the electron’s position is known, the less precisely are its speed and direction of motion known. Infinite precision is basically a nonphysical notion, but any desired finite degree of precision is, in principle, obtainable. The precision with which something can be measured is thus a good example of something that is potentially infinite, but never actually infinite.

But this still gives us an actual infinity in the world. For if our electron is located somewhere between zero and one, then each member of the following infinite collection is a possible outcome of a possible measurement:

.2 ± .1, .23 ± .01, .235 ± .001, .2356 ± .0001, …, .235608947 ± .000000001, …

Although infinite precision is impossible, an electron can be found to occupy any of the infinitely many points between zero and one whose distance from zero is a terminating decimal.

There are, however, some modern physical speculations that regard “space” and “time” as being abstractions which apply to our size level, but which become utterly meaningless out past the thirtieth decimal place. What would be there instead? Our old friend the *apeiron.* But even if we cannot really speak of infinitely many space locations, we might hope to find infinitely many sorts of particle.

It is sometimes thought that quantum mechanics *proves* that there is a smallest size of particle that could exist. This is not true. Quantum mechanics insists only that in order to “see” very small particles, we must use very energetic processes to look for them.

It is illuminating, after all this, to learn how the high-energy physicists actually go about finding new particles. The process is a little like finding stations on the radio by inching the dial back and forth until you hear music instead of static. One uses a particle accelerator in which collisions (between electrons and positrons) are continually taking place. The energy of the collision processes is varied by turning the voltage on the accelerator up and down. There is number R that measures the “particleness” of the reaction taking place. R can be thought of as being a little like the information parameter that enables you to tell whether you have found a station, even though the sound of music is no louder than the sound of the static. When an energy is found at which the graph of R versus energy has a sudden peak, then it is assumed that the energy in question is characteristic of the rest-mass of a new particle. This process is called “bump-hunting.” It is interesting to note that the sharper and narrower the peak, the more long-lived, and, thus, more “real” the particle is.

The question of whether or not matter is infinitely divisible may never be decided. For whenever an allegedly minimal particle is exhibited, there will be those who claim that if a high enough energy were available, the particle could be decomposed; and whenever someone wishes to claim that matter is infinitely divisible, there will be some smallest known particle which cannot be split. One is almost tempted to doubt if the question of the infinite divisibility of matter has any real meaning at all, particularly in view of the fact that such concepts as “matter” and “space” have no real meaning in the micro-world of quantum mechanics.

To return to something a little more concrete, let us consider the divisibility of our perceptual field. There is a limit to the subdivisions that this field can undergo. If two clicks happen close enough together in time, they cannot be distinguished; if a spot of ink is small enough, we can no longer see it. Hume makes much of this fact in his *Treatise of Human Nature* of 1739:

Put a spot of ink upon paper, fix your eye upon that spot, and retire to such a distance, that at last you lose sight of it; ’tis plain, that the moment before it vanish’d the image or impression was perfectly indivisible.23

The best way to understand Hume’s view of the world is to regard our space-time as being supplemented by an additional dimension of *scale.* To represent what I have in mind, let us forget about time and drop all the space dimensions but one. In Figure 24 I have drawn the space-scale continuum for a one-dimensional world. An individual’s perceptual field has a certain fixed size, as drawn; the field is made up of a certain finite number of slots or tiles—minimal perceptual units. In this model, the one-dimensional creature has two dimensions in which he can move his perceptual field. He can move to the left and right in space, and he can enlarge and contract his perceptual field. Rather than thinking of the field as enlarging and contracting, we think of the field moving up and down on the scale axis.

If the labelled objects (mountain, stone, speck of rock dust) occupy the appropriate regions of the space-scale continuum, then we can think of the ordinary perceptual level as being when the field is placed somewhere in the middle of the picture. At this perceptual level stones are visible, but one has neither enlarged one’s field of vision enough to see the mountain as a single object, nor contracted one’s attention enough to see the specks of dust on the rock. Notice that changing the size of one’s perceptual field amounts just to moving this field about in the space-scale continuum.

Hume takes perceptions as primary. Although he is often thought of as an empiricist, his is actually an extremely idealistic viewpoint. The perceptions are “out there”; one’s consciousness seems to move among them like a butterfly flitting from flower to flower.

One’s perceptual field has minimal elements, yet these minimal elements can be resolved into smaller elements by altering one’s field (by paying closer attention, using a telescope, or moving closer to the object in question). The only way to reconcile these two apparently contradictory aspects of our perceptual world is to view the world as a five-dimensional, space-time-scale continuum.

The question of the existence of an infinity in the small now becomes the question of whether or not the space-scale continuum drawn in Figure 24 extends *downward* indefinitely; similarly, the question of the existence of infinity in the large is the question of whether or not the continuum extends *upward* indefinitely.

I have long been interested in a curious trick that eliminates the infinity in the large and the infinity in the small without introducing any *absolute* perceptual minimum or maximum. This is simply the trick of bending the space-scale diagram into a tube, by turning the scale axis into a circle. Here the universe could consist of many galaxies, which consist of many star systems, which consist of many planets, which consist of many rocks, which consist of many molecules, which consist of many atoms, which consist of many elementary particles, which consist of many quarks and leptons, which consist of many darks, which could consist of many universes.24

A problem with the circular scale model is that if our universe is broken down far enough, one gets many universes, each of which will break down into many more universes. Are all of these universes the same? Perhaps, but then it would be hard to see how there could really be more than one object in the world. Another difficulty is that if there are many universes, each of which breaks up into many more universes, how can each of the component universes be one of the starting universes?

There is no problem if we have infinitely many universes. To illustrate this, I have drawn a picture of the simplest case: the case in which each universe is made up of two universes. We can see that 1 splits into 1 and 2, 2 splits into 3 and 4, 3 splits into 5 and 6, and in general *n* splits into 2*n* − 1 and 2*n*. We can continue splitting any given universe indefinitely, thus obtaining an infinite number of components in any bit of matter.

What is gained here is freedom from the belief that any size scale is intrinsically more basic or important or complex than any other size scale. Why waste time on the six o’clock news when you are no more nor less important than a galaxy or an atom? The point of this question is that one is often pressured to feel that the concerns of society or the world are more significant than one’s own immediate personal concerns. But this is based on the assumption that some sizes are in an absolute sense bigger than others, and it is this assumption that circular scale undermines.

#### CONCLUSION

In conclusion, note that it is entirely possible that our universe is in every sense finite. A toroidal space-time of the sort mentioned in the section on temporal infinities eliminates all infinities in the large; and if circular scale is introduced as in the section on infinities in the small, then there are no discrete infinities in the small. These finitizations can be accomplished smoothly: there need be no end of time, edge of space, or smallest particle.

But it is hard to believe that there would be only *one* of these totally finite universes. First, it is difficult to see how to apply circular scale unproblematically unless there are infinitely many universes; second, the principle of sufficient reason is violated if only *this* particular finite universe exists; and third, there is the feeling that the “space” in which our space-time is curved should be real.

In the section on spatial infinities it was pointed out that if, on the one hand, one repeatedly finitizes by replacing lines with circles, and if, on the other hand, one never accepts some particular finite *n*-verse as the end of the line–if, in other words, one thinks along the lines sketched in the last two paragraphs–then one is forced to conclude that space is infinite dimensional and that there are infinitely many objects in this cosmic space.

### INFINITIES IN THE MINDSCAPE

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In the last section I discussed some of the ways in which an actual infinity could physically arise. But there are things that are not physical. There are minds, thoughts, ideas, and forms. In this section we will see if any of these familiar nonphysical entities are actually infinite.

In order to appreciate the section at hand, it is necessary to keep an open mind on the question of whether or not mind equals brain, for if one assumes *a priori* that a thought is nothing more than a certain biochemical configuration in a certain finite region of matter, then (unless one has infinite divisibility of matter) it seems to follow automatically that infinite thoughts are impossible.

To cast a few preliminary doubts on the hypothesis that brain equals mind, let me quickly raise a few questions. Is what you thought yesterday still part of your mind? If you own and use an encyclopedia, are the facts in that encyclopedia part of your mind? Does a dream which you never remember really exist? How can you grasp a book as a whole, even though you only read it a word at a time? Would the truths of mathematics still exist if the universe disappeared? Did the Pythagorean theorem exist before Pythagoras? If three people see the same animal, we say the animal is real; what if three people see the same idea?

I think of consciousness as a point, an “eye,” that moves about in a sort of mental space. All thoughts are already there in this multi-dimensional space, which we might as well call the Mindscape. Our bodies move about in the physical space called the Universe; our consciousnesses move about in the mental space called the Mindscape.

Just as we all share the same Universe, we all share the same Mindscape. For just as you can physically occupy the same position in the Universe that anyone else does, you can, in principle, mentally occupy the same state of mind or position in the Mindscape that anyone else does. It is, of course, difficult to show someone exactly how to see things your way, but all of mankind’s cultural heritage attests that this is not impossible.

Just as a rock is already in the Universe, whether or not someone is handling it, an idea is already in the Mindscape, whether or not someone is thinking it. A person who does mathematical research, writes stories, or meditates is an explorer of the Mindscape in much the same way that Armstrong, Livingstone, or Cousteau are explorers of the physical features of our Universe. The rocks on the Moon were there before the lunar module landed; and all the possible thoughts are already out there in the Mindscape.

The mind of an individual would seem to be analogous to the room or to the neighborhood in which that person lives. One is never in touch with the whole Universe through one’s physical perceptions, and it is doubtful whether one’s mind is ever able to fill the entire Mindscape.

One last analogy. Note that there is always a certain region of physical space that only I can ordinarily know of—barring surgery, no one but me is in a position to assess the physical conditions obtaining within my stomach. In the same way, there is a certain part of the Mindscape that only I can ordinarily know of—unless I am to be greatly favored by the Muse, the feelings that pass over me when I think of my childhood will always remain private and inexpressible. Nevertheless, these almost ineffable feelings are part of the common Mindscape—they are simply difficult for anyone else to get to.

The point of all this is that just as the finiteness of our physical bodies does not imply that every physical object is finite, the finiteness of the number of cells in our brains does not mean that every mental object is finite.

Well … *are* there any infinite minds, thoughts, ideas, or forms or what have you in the Mindscape?

The most familiar candidate is the set *N* of all natural numbers. If I try to exhibit *N*, all I can really do is show you something like this: *N* = {1, 2, 3, …}. What the “…” stands for is something that is evident, yet basically inexpressible. The idea, of course, is that all of the natural numbers are to be collected together into a whole. Each of them would seem to exist individually in the Mindscape, and one would suppose that the set consisting of exactly the natural numbers would be in the Mindscape as well—one almost feels as if one can *see* it.

We might try to avoid the use of the “…” by saying something like this: “*N* is the set that has the following property: one is in *N*, and for any number *x* that is in *N, x* plus one is in *N* as well.” The trouble with this definition is that it does not uniquely single out one particular set. If, for instance, there were some infinitely large number *I*, and if *N** were the set consisting of all the numbers in *N* and all the numbers of the form *I* + *n* for some *n* in *N*, then *N** would satisfy the property that for every *x* in *N**, *x* plus one is in *N** as well … but *N** would be different from *N*.

We might try to get around this difficulty by saying that *N* is the *smallest set in the Mindscape* that has one in it, and that has *x* plus one whenever it has *x.* But, for reasons that I will begin to explain in the next section, the word “Mindscape” cannot be meaningfully used in a definition. The concept of “Mindscape” is too vast to be represented by any word or symbol.

If we try to avoid *this* difficulty by substituting some sort of finite description of the mental universe for the word “Mindscape,” then we get the same problem as before. By the classic work of the logician Thoralf Skolem, we know that for *any* finite description of *N* one might come up with, there will be a different set *N** that also satisfies the description. So it is quite literally true that what is really meant by the “…” is inexpressible.

Some thinkers have taken this to mean that there is, after all, no unique *N* in the Mindscape. This could be true. But one need not take this to mean that there are no infinite sets in the Mindscape: if there are many, many versions of the set of natural numbers, then there are many, many infinite sets. However, it is normally more desirable to assume that there is a simple unique *N* in the Mindscape, just as it is simpler to assume that there is only one universe instead of a whole slew of “parallel worlds.”

I might note here that if time is indeed infinite, then just as we can indicate Earth by saying, “this planet,” we could indicate our *N* by saying, “the number of seconds left in this time.” This is, in fact, what people do when they attempt to define *N* by saying, “*N* is what you get if you start with one and keep adding ones forever.”

If infinite forms are actually out there in the Mindscape, then maybe we can, by some strange trick of mental perspective, *see* some of these forms. The philosopher Josiah Royce maintained that a person’s mental image of his own mind must be infinite.25 His reason is that one’s image of one’s own mind is itself an item present in the mind. So the image includes an image that includes an image, and so on. This infinite regress can be nicely visualized by imagining a United States in which a vast and fanatically accurate scale model of the country occupies most of the Midwest. The scale model, being absolutely accurate, includes a copy of the scale model, etc. This regress is occasionally used to make a striking label for a commercial product. The old can of Pet Milk, for instance, bore a picture of a can of Pet Milk, which bore a picture of a can of Pet Milk, etc.

In a physical situation we would probably never actually be able to finish making such a label in all its infinite detail. But this is not to say that no such label or country-plus-scale-model could *exist.* There would be no problem, if matter were infinitely divisible. (If scale is indeed circular, then everything is, in a sense, already an object of this nature!)

There is certainly no reason why a nonphysical mind should not be infinite; and Royce’s point is that if you believe that one of the things present in your mind is a perfect image of this mind and its contents, then your mind *is* infinite. One might try to avoid this conclusion by adopting a circular scale attitude and insisting that there is no difference between the mind and the mind’s image of itself, so that the allegedly infinite set of thoughts {image of the mind, image of the image, image of the image of the image, …} is really the same as the set {mind, mind, mind, …}, which is just a set with one member: {mind}.

I would like to discuss this a bit more, but first let me formally introduce some of the apparatus of set theory. In Cantor’s words, “A set is a Many that allows itself to be thought of as a One.”26 A set is usually given as a pair of curly brackets enclosing some description of the contents of the set. It is easiest to think of the curly brackets as a thought balloon. Thus the set {1, 2} is the unity obtained by taking the multiplicity consisting of the numbers 1 and 2 and treating this multiplicity as a unity. That is, we can think of the set {1, 2} as being represented by a thought balloon that has 1 in it and 2 in it.

Of particular interest in set theory is the *empty set*, ∅. ∅ is the One obtained by taking together … nothing. If we write out ∅ in the ordinary way we get { }, which I have drawn as an empty thought balloon.

More and more complicated sets can be built up using only the brackets in various arrangements. Thus we have the set {{ }} depicted in Figure 28B, and we could equally well form {{ }, {{ }}, {{ }, {{ }}}} which is how the number 3 is usually represented in terms of pure sets. (See Figure 29.)

Now let’s get back to the question of whether or not a mind that has a perfect self-image is infinite. Really to get down to the bare bones, say that we have a mind or label or set *M* such that the only member of *M* is *M*. That is, *M* = {*M*}. Now, if we change this equation by replacing the *M* on the right by {*M*} then we get *M* = {{*M*}}. If we could continue replacing *M* by {*M*} forever, we would wind up with *M* = {{{{{… …}}}}} · This could actually be a definition of an *M* whose only member is itself, for note that placing another pair of brackets around {{{{{… …}}}}} changes nothing. In plain English, *M* is the set whose only member is the set whose only member is …

But if the only member of *M* is indeed *M* itself (rather than a copy of *M*), then *M* really only has one element. It is just that if we try to describe this element by using brackets we get an infinite description. We call thoughts like *M self-representative.* Whether or not such an *M* is to be regarded as infinite depends on whether you experience the *M* subjectively (in the way you experience your own mind), or objectively (as a feature of the Mindscape that is to be precisely described in the language of set theory).

Set theory is, indeed, the science of the Mindscape. *A set is the form of a possible thought.* Set theory enables us to put various facts about the Mindscape into one framework in the same way that the atomic theory of matter provides a framework in which the diverse physical and chemical qualities of matter can be simultaneously accommodated.

Before the atomic theory of matter, such phenomena as melting and burning, rusting and freezing were regarded as qualitatively different. Once a good atomic theory was developed, however, all of these phenomena could be thought of in more or less the same way. The notion of set was consciously introduced only at the turn of the century. Before long, it became evident that all of the objects that mathematicians discuss–functions, graphs, integrals, groups, spaces, relations, sequences–all can be represented as sets. One can go so far as to say that mathematics is the study of certain features of the universe of set theory.

The universe of set theory is closely bound up with the Mindscape—one can, perhaps, think of the former as a sort of blueprint of the latter. A set is obtained when we take a thought and abstract from it all the emotive content, keeping only the abstract relational structure. A set is the form of a possible thought. So the question of whether or not there are any infinite entities in the Mindscape is really equivalent to the question of whether or not there are any infinite sets.

According to set theorists, there certainly *are* infinite sets. Indeed, there is to be an endless hierarchy of infinities: the set of natural numbers, the set of all sets of natural numbers, the set of all sets of sets of natural numbers, etc. Each member of this sequence can be shown to be of an infinity greater than that of the earlier members. In modern set theory there is a whole field of study called large cardinals, whose specialists study a dizzying array of higher and higher infinities.

But many mathematicians and philosophers do not go along with the set theorists. The traditional *finitist* viewpoint is still with us. According to the finitists, there is nothing that is infinite, in heaven or on earth.

Those who assert that infinite sets of every size have a secure existence in the Mindscape are usually called *Platonists.* This name is a bit inapt, since Plato did not believe in infinity; but he *did* believe in the existence of ideas independent of thinkers, and it is for this aspect of his thought that the Platonists are named.

It is not likely that the finitist vs. Platonist debate will ever be concluded. On the one hand, it is probably impossible to meet the demands of a finitist who says that he will believe in infinity only if he is *shown* an infinite set right now; on the other hand, the notion of infinite sets appears to be logically consistent, so the finitist can never prove that infinite sets do not exist.

I incline towards Platonism; but if you are stubborn enough, how can I possibly convince you that infinite things are real? All I can do, after all, is to make a finite number of marks on a finite number of sheets of paper. If you are truly committed to disbelief in the infinite, then you will not be satisfied by anything less than my simultaneously exhibiting each member of some infinite set … and whenever I claim that I have done so, you will triumphantly point at the finiteness of the number of marks on paper which I have really shown you.

In pre-Cantorian times finitists sometimes thought that they had proved the impossibility of actually infinite sets. These proofs, however, were always fallacious. Such proofs usually deal with some particular property *P* of numbers that each natural number happens to enjoy. *P* might be the property of being odd or even, having an immediate predecessor, being the sum of finitely many units, or being greater than any predecessor. The false proof that no infinite numbers exist then takes the form: “Every number has property *P.* If *x* is an infinite number, then *x* cannot have property *P*. Therefore no infinite numbers can exist.” The fallacy in such a circular proof is that when it is asserted that “every number has property *P*,” it is being quietly assumed that anything that fails to have property *P* does not exist.

But, of course, one cannot assume that the infinite sets must have certain properties before one has ever looked at them! Galileo’s paradox, for example, showed that an infinite set can be put into a one-to-one correspondence with a proper subset of itself. Had we assumed in advance that no set could be put into a one-to-one correspondence with a proper subset of itself, then we would have had a proof that no infinite set can exist. But such an assumption is *totally unwarranted;* indeed, to make such an assumption is essentially to assume in advance that every set is finite … which does not make for a very productive debate.

But are we quite sure that the finitists will never come up with some *valid* proof that the notion of infinite sets is incoherent and fundamentally meaningless? A Platonist would answer that yes, he is sure that there is no inconsistency in the theory of infinite sets. He is sure of this because the theory in question is a description of certain features of the Mindscape that “anyone can see.”

But the finitist can still hope. There is a curious proof, discovered by Kurt Gödel in 1930, that the consistency of set theory cannot be finitely proved. The time will never come when the finitist is absolutely forced to admit that it is safe to talk about infinite sets.

In mathematics no other subject has led to more polemics than the issue of the existence or nonexistence of mathematical infinities. We will return to some of these polemics in the last chapter. For now, let us reprint Cantor’s opening salvo in the modern phase of this age-old debate:

The fear of infinity is a form of myopia that destroys the possibility of seeing the actual infinite, even though it in its highest form has created and sustains us, and in its secondary transfinite forms occurs all around us and even inhabits our minds.27

Strong words! But what does Cantor mean when he says that the highest form of infinity created us? Read on!

### THE ABSOLUTE INFINITE

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There is a certain type of nonphysical entity that was not discussed in the last section. God, the Cosmos, the Mindscape, and the class V of all sets—all of these are versions of what philosophers call the Absolute. The word “Absolute” is used here in the sense of “non-relative, non-subjective.” An Absolute exists by itself, and in the highest possible degree of completeness.

As I mentioned earlier, Plotinus held that the One could not be limited in any sense. As Aquinas, the quintessential theologian, says: “The notion of form is most fully realized in existence itself. And in God existence is not acquired by anything, but God is existence itself subsistent. It is clear, then, that God himself is both limitless and perfect.”28

The limitlessness of God is expressed in a form closer to the mathematical infinite by St. Gregory: “No matter how far our mind may have progressed in the contemplation of God, it does not attain to what He is, but to what is beneath Him.”29 We have here the rudiments of the infinite dialectic process that takes place if we systematically try to build up an image of the whole Mindscape.

Suppose that I want to add thought after thought to my mind until my mind fills the whole Mindscape. Whenever I make an attempt at this, I am collecting together a group of thoughts into a single thought *T*. Now, when I become conscious of my state of mind *T*, I realize that this is a new thought that I had not yet accounted for … so I improve my image of the Mindscape by passing to the thought that includes all the elements of *T* plus *T itself*, viewed objectively.

This is a dialectic process in the sense that the thetic component is one’s instantaneous unconscious image of the Absolute, the antithetic component is the conscious formalization of this image, and the synthetic component is the formation of a new unconscious image of the Absolute that incorporates one’s earlier images and the awareness that they are inadequate.30

This process is most clearly understood if we start with nothing at all, as in the cartoon strip of Wheelie Willie in Figure 32. (Wheelie Willie is a character whose adventures I occasionally used to draw for the *Rutgers Daily Targum* when I was in graduate school there.) Notice that in each of the shifts, what takes place is that Wheelie Willie forms a thought that has as its members the members of the last thought plus the last thought itself. Looked at another way, the thought at each stage has all of the previous stages as components.

If we call the *n*th thought *T _{n}*, we can define

*T*in two ways. On the one hand, we can use an inductive definition:

_{n}*T*

_{0}= ∅ and

*T*

_{n+1}=

*T*∪ {

_{n}*T*}, where for any sets

_{n}*A*and

*B, A*∪

*B*means the set of all the sets that are members of

*A*or of

*B.*On the other hand, we can use a different sort of inductive definition:

*T*= {

_{n}*T*}, which means “

_{m}:m < n*T*is the set of all

_{n}*T*such that

_{m}*m*is less than

*n*.”

Some readers may have asked themselves if the thought *T* plus “*T*” really has to be different from the thought *T*. And the answer is, not always. In the last section we were looking at a mind, *M*, which has *M* as one of its components. Such an *M* is already fully self-aware, and *M* plus “*M*” is no different from *M*. In terms of sets, *M* ∪ {*M*} = *M*.

It would seem, in particular, that God should be able to form a precise mental image of Himself. Insofar as the Mindscape is God’s mind, what I am saying is that one of the objects in the Mindscape should be the Mindscape itself. That is, the Mindscape is an *M* that has *M* as one of its members. Now, any object in the Mindscape is, in principle, something that one can perceive through one’s consciousness. So it would seem to be possible for our minds actually to attain a vision of God or of the whole Mindscape.

Now this seems to contradict St. Gregory’s dictum and the general feeling that the Absolute is unknowable. But there are two *kinds* of knowing: the rational and the mystical.

If I know something rationally, then I have some thought that is built up from simpler thoughts, which are in turn built up from still simpler thoughts. This regress is not infinite, but goes only through some finite number of stages before certain simple and unanalyzable perceptions and ideas are reached. My idea of “house” consists of a collection of ideas, each one of which represents a certain type of house (e.g., my house, brick house, hovel). Each idea of a type of house consists of ideas of various components and functions (doors, windows, shelter), which can in turn be explicated in terms of certain simple ideas (walking, vision, warmth).

When I communicate a rational thought, what I do first is to show what the components of my thought are, and then to show how the components fit together. If one of the components of the final thought were to be the final thought itself, then this rational communication would be blocked by an infinite regress. To explain the thought, I would first have to explain the thought. I could not finish unless I had already finished.

In terms of rational thoughts, the Absolute is unthinkable. There is no non-circular way to reach it from below. Any real knowledge of the Absolute must be mystical, if indeed such a thing as mystical knowledge is possible.

Mathematics and philosophy do not normally have a great deal to say about the mystical way of knowing things. Mystically speaking, it is possible to experience a direct vision of the whole Mindscape. This vision cannot be rationally communicated for the reasons just outlined. Of course, it is possible to communicate mystical knowledge in an indirect way, for example, by advocating that a person prepare his or her mind through carrying out some physical or spiritual exercises. But, ultimately, mystical knowledge is attained all at once or not at all. There is no gradual path by which to build up an *M* that has *M* as one of its elements.

Even if full knowledge of the Absolute is only possible through mysticism, it is still possible and worthwhile to discuss *partial* knowledge of the Absolute rationally. A significant thing about the Mindscape and the other Absolutes is that they are actually infinite. Indeed, in 1887 Cantor’s friend, Richard Dedekind, published a proof that the Mindscape is infinite, where Dedekind’s word for Mindscape was *Gedankenwelt*, meaning thought-world.31

Dedekind’s argument for the infinitude of the Mindscape was that if *s* is a thought, then so is “*s* is a possible thought,” so that if *s* is some rational non-self-representative thought, then each member of the infinite sequence {*s, s* is a possible thought, *s* is a possible thought is a possible thought, …} will be in the Mindscape, which must, therefore, be infinite.

A very similar argument proves that the class of all sets is infinite. The class of all sets is normally called V, or Cantor’s Absolute. We can use the Wheelie Willie sequence of sets to see that there are infinitely many different sets in V.

Dedekind modelled his argument after an argument that appears in Bernard Bolzano’s *Paradoxes of the Infinite* (ca. 1840):

“The class of all true propositions is easily seen to be infinite. For if we fix our attention upon any truth taken at random …, and label it *A*, we find that the proposition conveyed by the words ‘*A* is true’ is distinct from the proposition *A* itself…”32

So we can see that the Mindscape, the class of all sets, and the class of all true propositions are all infinite. Does this guarantee that infinite objects exist? Not really. For a case can be made for the pluralist claim that the Mindscape, the class of all sets, and the class of all true propositions do *not* exist as objects, as unities, as finished things.

In more familiar terms, it is not hard to prove that God is infinite … but what if you don’t believe that God exists? It may seem hard to doubt that the more impersonal Absolutes–such as “everything,” or the Mindscape–exist, but there are those who do doubt this. The issue under consideration is a version of the old philosophical problem of the One and the Many. What is being asked is whether the cosmos exists as an organic One, or merely as a Many with no essential coherence. It is certainly true that the Mindscape, for instance, does not exist as a single rational thought. For if the Mindscape is a One, then it is a member of itself, and thus can only be known through a flash of mystical vision. No rational thought is a member of itself, so no rational thought could tie the Mindscape into a One.

Normally the word “set” is restricted by definition to apply only to collections that are not members of themselves. Under this use of the word, the class V of all sets cannot be a set, for if it were, we would have a set V such that V is a member of itself. So V becomes a collection that can never be formed into a One.

Suppose that we do not believe in circular scale and assume that any physical thing is not a part or component of itself. Is the Cosmos, the collection of all physical things, a thing? If it is, then it has to be a component of itself, which we do not allow. So the Cosmos is not a thing, but only a Many that can never be a One.

There is a highly relevant passage in a letter Cantor wrote to Dedekind in 1905:

“A multiplicity can be such that the assumption that *all* its elements ‘are together’ leads to a contradiction, so that it is impossible to conceive of the multiplicity as a unity, as ‘one finished thing.’ Such multiplicities I call *absolutely infinite* or *inconsistent multiplicities.* As we can readily see, the ‘totality of everything thinkable,’ for example, is such a multiplicity…”33

Again, the reason that it would be a contradiction if the collection of all rational thoughts were a rational thought *T* is that then *T* would be a member of itself, violating the rationality of *T* (where “rational” means non-self-representative). The upshot of all this is that God, the Mindscape, the class of all sets, and the class of all true propositions all seem to be infinite, but it is at least possible to question whether any of these Absolutes exists as a single entity. Certainly they do not exist as entities that can be fully grasped by the rational mind.

### CONNECTIONS

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In this section I would like to explore some of the connections between the various sorts of infinities that have been discussed.34 In his 1887 essay, “Contributions to the Study of the Transfinite,” Cantor quotes a passage from Aquinas’s *Summa* and states repeatedly that in this passage appear the only two really significant objections that have ever been raised against the actual infinite.35 Let us examine this quote from Aquinas here, reproducing Cantor’s italics:

The existence of an actually infinite multitude is impossible. 1) For any set of things one considers must be a *specific* set. *And sets of things are specified by the number of things in them. Now no number is infinite*, for number results from counting through a set in units. So no set of things can actually be inherently unlimited, nor can it happen to be unlimited. 2) Again, every set of things existing in the world has been created, and anything created is *subject to some definite purpose* of its creator, *for causes never act to no purpose.* All created things must be subject therefore to definite enumeration. Thus even a number of things that happens to be unlimited cannot actually exist.36

It seems clear that Aquinas’s first point is that an infinite set can occur only if infinite numbers exist, and he does not believe that infinite numbers exist. Cantor’s theory of transfinite numbers stands as the only adequate response to this objection. For many years, it was believed that the notion of actually infinite numbers was fundamentally incoherent. It was only with the birth of Cantor’s theory in the late 1800s that a consistent and reasonable theory of infinite, or transfinite, numbers was developed. As Cantor remarks in his discussion of Aquinas’s objection, this objection against the existence of actually infinite collections is to be met *positively* by exhibiting a theory of infinite numbers.

It is not so obvious what Acquinas’s second point might be. It might be taken to be simply a variation on the first point. Under this reading, the first point says that any set must have a number of cardinality, but all numbers are finite; and the second point says that any set must have a purpose or significance, but any definite purpose is finite. If this is indeed Aquinas’s meaning, then we can say that once again the Cantorian theory of infinite sets provides a positive rebuttal.

Aquinas’s whole view of the infinite is not really tenable, for he held that God is infinite, but that no created thing is infinite. This contradicts a widely accepted principle known as the *Reflection Principle.* The Reflection Principle as formulated in set theory goes as follows: every conceivable property that is enjoyed by V is also enjoyed by some set. (Recall here that V is Cantor’s Absolute, the class of all sets.) Philosophically it would run: every conceivable property of the Absolute is shared by some lesser entity; or, every conceivable property of the Mindscape is also a property of some possible thought.

The motivation behind the Reflection Principle is that the Absolute should be totally inconceivable. Now, if there is some conceivable property *P* such that the Absolute is the only thing having property *P*, then I can conceive of the Absolute as “the only thing with property *P*.” The Reflection Principle prevents this from happening by asserting that whenever I conceive of some very powerful property *P*, then the first thing I come up with that satisfies *P* will *not* be the Absolute, but will instead be some smallish rational thought that just happens to reflect the facet of the Absolute that is expressed by saying it has property *P*.

Let me give an example of a Reflection Principle argument. *For every thought S in the Mindscape, the thought “S is a possible thought” is also a thought in the Mindscape.* By Reflection there must, therefore, be some thought *W* such that For *every thought S in W, the thought “S is a possible thought” is also in W.* This *W* reflects, or shares, the italicized property of the Mindscape. But note now that this *W* must be infinite. So an infinite thought exists.

Again, it is true that each of the Wheelie Willie sets *T _{n}* is a member of V. By the Reflection Principle there must, therefore, be some set

*N*such that each of the Wheelie Willie sets

*T*is a member of

_{n}*N*. Therefore an infinite set

*N*exists.

The point I wish to make is that if one accepts the existence of any of the various infinite Absolutes, then one is fairly well committed to accepting the existence of infinite thoughts and sets. For to deny the Reflection Principle is practically to assert that the Absolute can be finitely described, which is most unreasonable.

The passage from St. Augustine that I referred to earlier contains a kind of Reflection Principle argument for the reality of the set *N* of all natural numbers. In that passage Augustine argues that God must already know each and every natural number and that he even knows “infiniteness” in the form of all the natural numbers taken at once—for otherwise the set of natural numbers would exhaust his abilities. God, according to Augustine, must lie *beyond* the set of natural numbers.

To summarize the points in this chapter:

1. The infinite normally inspires such feelings of helplessness, fatility, and despair that the natural human impulse is to reject it out of hand.

2. There are, however, no conclusive proofs that everything is finite; and the question of whether or not anything infinite exists remains as an open, almost empirical problem.

3. There are various sorts of physical infinites that could actually exist: infinite time, infinitely large space, infinite dimensional space, infinitely continuous space, and infinitely divisible matter. Each of these infinites is, in principle, avoidable; whether or not our Cosmos actually does avoid infinities remains to be seen.

4. In Cantor’s set theory we have a great number of infinite sets. This simple and coherent theory of the infinite provides a logical framework in which to discuss infinities. Moreover, if we feel that the things that mathematicians discuss are real, then we can conclude that actually infinite things exist.

5. Attempts to analyze the phenomenon of consciousness and self-awareness rationally appear to lead to infinite regresses. This seems to indicate that consciousness is essentially infinite.

6. The Absolute is certainly infinite. So one must either deny the reality of the Absolute or accept the existence of at least one infinity.

7. According to the Reflection Principle, once one has an infinite Absolute, one must also have many conceivable infinities as well.

### PUZZLES AND PARADOXES

(*Answers on Page 295*)

1. It is sometimes said that if infinitely many planets existed, then every possible planet would have to exist, including, for instance, a planet exactly like Earth, except with unicorns. Is this necessarily true?

2. Consider a very durable ceiling lamp that has an on-off pull string. Say that the string is to be pulled at noon every day, for the rest of time. If the lamp starts out off, will it be on or off after an infinite number of days have passed?

3. For each observer *O*, there is some fixed upper bound *N _{o}* to the number of stars that

*O*can physically see. Therefore, for each observer the universe is finite. Does this imply that the universe is finite?

4. “I have five fingers on my left hand,” means the same thing as, “When I count up all the fingers on my left hand, the last number I say is *five.*” What might “I have ∞ fingers on my left hand” mean?

5. Suppose that we find an infinite number *I* that is the largest possible number. But now, what about *I* + 1?

6. In the little-known field of “enumerative geometry,” it is said that there are ∞ points on a line and ∞2 points in a plane. There are said to be ∞2 *lines* in the plane as well: “To get the correct number ∞2 of straight lines in the plane, we must divide the number ∞4 of pairs of points in the plane by the number ∞2 of pairs of points on a straight line.”37 How many circles should there be in the plane? How many ellipses?

7. Can you prove, without circularity, that seven is a finite number?

8. The universe has lasted about 1010 years since the Big Bang. There are about 3 × 107 seconds in a year. According to quantum mechanics, the usual conception of continuous time does not extend to intervals shorter than 5 × 10−44 seconds, so we might think of this unit as being a kind of “instant,” faster than which nothing can happen. How many “instants” of time does that come to so far? Is it reasonable to argue that larger numbers, such as 10100, do not yet exist?

9. Say that the space we live in is infinitely large. Consider an infinite line *L* contained in our space. *L* is infinity yards long, and *L* is infinity feet long. But since each yard is three feet, *L* is also three-times-infinity feet long. How can infinity equal three times infinity?38

10. Here is an example of an infinite regress. Suppose that some person wishes to prepare a text in which every appearance of the letters “man” is replaced by the letters “woman.” If this is rigidly adhered to, then “man and woman” becomes “woman and wowoman,” then “wowoman and wowowoman,” and so on. what do you reach in the limit?

## CHAPTER TWO

## ALL THE NUMBERS

In this chapter we will begin by tracing the development of the familiar real number system with its infinity of irrational numbers. Once one has accepted irrational numbers, there is really no reason not to accept infinitely large or transfinite numbers. So the second section of the chapter will be devoted to the transfinite ordinals and cardinals. The ordinals form a gappy number sequence somewhat like the natural numbers. It is a natural move to fill in these gaps as densely as possible, just as one fills in the space between, for instance, two and three with rational and real numbers. If we fill in as much as possible we end up with what might be called an *absolutely continuous* ordering. In the section on infinitesmals and surreal numbers I will present some examples of such orderings, any one of which can be viewed as comprising “all the numbers” (including the infinitesmals). In the final section of this chapter I will return, once again, to the question of whether the infinitely big and infinitely small numbers have any real existence, physically or otherwise.

### FROM PYTHAGOREANISM TO CANTORISM

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Pythagoras lived in Greece and in Italy in the sixth century B.C. He is an extremely shadowy and ambivalent figure. On the one hand, he was a wizard, the shamanistic leader of a religious sect. On the other, he has frequently been credited with bringing about the birth of modern mathematics and mathematical physics.

The sect of Pythagoreans is best known for their belief in *metempsychosis*, or reincarnation. They believed that there is one cosmic mind or soul, that you are alive because a small piece of this soul is imprisoned in your body, and that the bit of soul that animates you will animate many other bodies before returning to full unity with the one big soul. The Pythagoreans adhered to a great number of rules and taboos (never look back when crossing a border, always put the right shoe on first, never pick up food that drops from the table), apparently in an effort to bring themselves into a closer harmony with the cosmos. Presumably it was hoped that if in the course of your lifetime you could bring yourself into a close enough relationship with the One, then when your body died, the soul that vivified it might be able to return to the source instead of being forced into another body.

Pythagoras was said to be able to remember several of his previous lives, and he was believed to have many other supernatural powers as well. There is a whole series of ancient miracle tales about Pythagoras, such as stories that he was once seen in widely separated places at the same time, and that once when he was crossing a river it hailed him in an audible voice saying, “Greetings, Pythagoras.”

Part and parcel of the Pythagorean religious beliefs were a number of numerological notions. There was a feeling that the essential nature of the cosmos was somehow numerical, with certain numbers seeming to embody particular abstract concepts. The Pythagoreans made the following identifications: 1 was mind (the One); 2 was opinion (the first moving away from unity); 3 represented wholeness (beginning, middle, and end); 4 was justice (a “square” deal); 5 stood for marriage (since 5 = 2 + 3, and even numbers were regarded as female, odd as male). Under a later system the numbers one through four were identified with the point, line, plane, and solid, respectively.

The number ten was singled out for special attention and was said to symbolize perfection. One reason for this is obvious: people have ten fingers, and most of our systems of numeration are based on the number ten. But a more important reason for the importance of the number ten is that 10 = 1 + 2 + 3 + 4, and the numbers one through four and their interrelations were regarded as primary. This fact about 10 was represented by the Pythagorean *tetractys* depicted in Figure 36. A Pythagorean would feel right at home in a bowling alley, ritually building and destroying the *tetractys* with a sphere punctuated by a triad of holes, and recording his progress with a series of numbers inscribed in squares.

The Pythagoreans assumed that since ten was so important, there should be ten heavenly bodies. At the time there were only nine known celestial objects (not counting the stars), so the Pythagoreans postulated the existence of a counter-Earth that is never seen, because it is always on the opposite side of the sun.

It is interesting to realize that this *type* of argument is the stock in trade of modern mathematical physics. For example, a three-dimensional chart of all the known elementary particles is drawn up. The chart looks, let us say, like a regular dodecahedron with one corner missing. It would look prettier, more symmetric, if there were an additional particle with such and such characteristics to fill the missing corner, so the physicists postulate the existence of such a particle. The surprising thing is that, often as not, such an argument turns out to be correct: a particle with exactly the predicted properties is discovered.

The fact is that *a priori* mathematical considerations can lead to empirically determined physical truths. The structure of the physical universe is deeply related to the structure of the mathematical universe. The Pythagoreans were aware of examples of this relationship, having observed, for instance, that if the lengths of two stretched strings are in a simple numerical ratio (such as 2:1 or 3:2 or 4:1), then the notes produced by plucking the strings are consonant.

The conclusion that the Pythagoreans drew was, according to Aristotle, “that the elements of numbers are the elements of things, and that the whole heaven is a harmony and a number.” Again, Aristotle states that the Pythagoreans “considered number as the substance of all things.”1

This sort of viewpoint is not uncongenial to the modern scientist, for whom any phenomenon can be expressed in terms of numbers, vectors, functions, operators, groups, and the like. If one believes that the universe is basically all form and no content and that the forms that arise in nature all admit of mathematical representation, then one can reasonably conclude that anything that exists is ultimately a mathematical object.

Take my right shoe, for example. I can, of course, state the size, count the number of eyelets, or determine the weight in grams. But even independently of my efforts, the shape exists mathematically as the set of coordinates of points that happen to lie within the substance of the shoe; and the color of the shoe is precisely specified by a function giving the wave lengths of light reflected at each point of the surface of the shoe. As for the actual particles that make up the shoe, they may very well be nothing more than small irregularities in the curvature of space-time. So it is not really so odd to believe with the Pythagoreans that ultimate reality is precise mathemetical form.

LIMITED | UNLIMITED |

ONE | MANY |

REST | MOTION |

STRAIGHT | CROOKED |

GOOD | BAD |

So far, so good. But the story gets more interesting. The Pythagoreans did not believe in infinite forms. They are credited with the creation of a “table of opposites,” which I have partially reprinted. Looking at this table, it is evident that the Pythagoreans would not have been big fans of infinity. In the original Greek they have *apeiron* for “unlimited.”

Now, if i) everything is a mathematical form and ii) nothing infinite exists, then everything is basically either a natural number or some relationship between a few natural numbers. Note that you have to give up either i) or ii) if it can be proved that there is some feature of the world that cannot, even in principle, be fully represented by a finite number of natural numbers.

Imagine the coast of Southern Italy. You’re out on the water … brilliant, ultramarine water and dry, rugged rocks. Pythagoras himself is aboard the boat on this outing of the brotherhood. There’s been a lot of hassles with the locals, and everything is finally mellow, kind of *merged* out here on the water. Pythagoras is sitting on the deck talking with Hippasus, a guy in his thirties who laughs a lot. Hippasus is scratching lines on the smooth deck with a knife, showing Pythagoras and some of the others a construction he’s been fooling with. It goes something like this.

Once a square and the diagonal of the square have been drawn, one can ask about the ratio *d:s* of the length of the diagonal to the length of the side. If all things are expressible in terms of natural numbers, then one would expect that there are two natural numbers *m* and *n* such that *d:s::m:n. But it can be conclusively proved that no such natural numbers m and n exist.* The ratio *d:s* is irrational, nameless, *apeiron.*

In the mental movie we were just watching, Hippasus is letting Pythagoras in on this. One version of the movie’s ending is that when the Pythagoreans returned from their sail, it turned out that Hippasus had “drowned at sea”!

Given the Pythagorean theorem, which states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the sides, we can see that *d*2 = *s*2 + *s*2, so *d*2 = 2*s*2, so *d2/s2* = 2, so the ratio In modern terms, what the Pythagoreans learned was that is an *irrational* number. But as far as they were concerned, they had discovered that there was a physical relationship, the ratio *d:s*, which was not representable in terms of numbers. Since they did not recognize the existence of ratios other than the natural number ratios, the ratio *d:s* was called *alogos*, meaning “inexpressible.” It was also called *arratos*, meaning “not having a ratio.”

It is interesting to see how one might go about trying to find a representation of as a fraction, or ratio, of natural numbers. This amounts to the problem of finding an *n* such that for some *m, m*2/*n*2 = 2*n*2/*n*2. In the table below, I have sketched the beginning of a search for such an *n.* The curious thing is that we can say with certainty that this search must remain forever fruitless. The proof is covered in almost every survey course of mathematics.

For the Greeks there were two kinds of magnitudes: discrete and continuous. Discrete magnitudes could be counted, set into correspondence with natural numbers that were sometimes visualized as patterns of dots. But continuous magnitudes simply did not correspond to any *number* at all. Just as we can add and multiply numbers, we can manipulate continuous magnitudes by means of the techniques called geometrical algebra. The Greeks developed these techniques to the point where they could, in effect, solve most quadratic equations involving continuous magnitudes.

Consider, for example, the geometric technique for finding the mean proportional between two lengths *a* and *b.* That is, given line segments *a* and *b*, we wish to find a line segment *m* such that *a:m::m:b.* The construction of *m* is as follows.

1. Put *a* and *b* end to end, forming *AC*.

2. Construct a semicircle having *AC* as diameter.

3. Erect a perpendicular to *AC* at *B*, meeting the semicircle at *D*, let *AD* have length *m.*

4. *a:m::m:b* because triangle *ABD* is similar to triangle *DBC.*

In modern terms we would say that *m* is a solution of the equation *a/x* = *x/b, a · b = x*2, or The fact that we can solve the second to last equation is expressed geometrically in Proposition 14 of Book II of Euclid’s *Elements*, which says in part that, “It is possible to construct a square equal [in area] to any given rectangle.”2

Notice how differently the problem of finding the mean proportional is treated today. Since i) we think of there being a real number corresponding to every length, and since ii) we have extended all of the usual operations, such as multiplication and square root, to the real numbers, we are able to assert that i) any given line segments have some real number lengths *a* and *b* and that ii) there is a real number

What are these protean real numbers of ours? In general, a non-negative real number has the form *n.r*_{1}*r*_{2}*r*_{3}*r*_{4}*r*_{5} …, where *n* is a natural number, and each of the *r*_{i} is one of the digits 0 through 9. The interesting thing about these “real” numbers is that they are, in point of fact, very *ideal* objects. The string of digits to the right of the decimal place is infinite. Strictly speaking, a real number can never be completely written out.

Of course, some real numbers, such as 25.000 … or 3.123123123 …, eventually begin repeating themselves. For convenience we write these numbers as follows: where it is understood that the string of digits under the bar is repeated over and over. There is an interesting little theorem about repeating decimals. To state this theorem, we must keep in mind that a real number is *rational* if it is equal to some fraction, such as

**Theorem:** A real number *r* is rational if and only if it has a repeating decimal expansion.

Instead of giving a formal proof of the theorem, let’s just see an example of how each direction works. First, imagine that you have a real number *r* that is equal to the fraction In order to get the decimal expansion of *r*, we begin dividing 7 into 2. I have circled the successive remainders that occur. Notice that i) when you are dividing by 7 the remainder is always one of the natural numbers 0 through 6, and ii) if the same remainder occurs twice, then the decimal starts repeating, since the same sequence of actions will follow.

Second, imagine that there is a real number *r* that has the repeating decimal expansion .123. Now, *r* has an infinite number of blocks of “123” to the right of the decimal, so if we move one of these blocks to the left of the decimal, by taking 1000 *r*, then there will still be an infinite number of “123” blocks to the right of the decimal.

So, in the indicated calculation, if we subtract *r* from 1000 *r*, then there is nothing remaining on the right of the decimal.

It is satisfying to see that the two finite ways of describing a real number–as a fraction, or as a repeating decimal-coincide. Given that we have proved that is irrational, we can be sure that the decimal expansion of *never repeats.* That is, when we write we do not have any simple way of describing the pattern that the “…” stands for.

We can also use the theorem in the reverse way. That is, we can artificially construct a non-repeating decimal and be sure that this represents an irrational real number. There is, for instance, the artificial number of Liouville, .010010001000010000010000001 …, where the building principle of steadily increasing the number of zeros between the ones guarantees that the number never repeats itself. A different sort of non-repeating decimal can be obtained by sticking together all of the natural numbers to get the “number number,” .1234567891011121314151617181920212223….

But how can we be sure that such artificial decimal expansions are really numbers? What exactly is *meant* by .12345…? The understanding is that .12345 … stands for the infinite series or sum, It is easy to visualize a geometric interpretation.

One is tempted just to say that a given real number, such as .12345 …, is really to be thought of as a point on the idealized real number line. The problem with this approach is that one has not really explained where the “real number line” comes from. The real number line is basically something to be found with certainty only in the Mindscape; there is no reason to assume in advance that our physical space is filled with copies of the real number line.

This problem was finally dealt with only about one hundred years ago, by our old friends Cantor and Dedekind. Cantor basically defined a real number simply as an infinite sequence of digits, just as was done above. The original element of his approach was that one does not act as if the limit or sum of the infinite series expressed by a real number is anything *other than* or *external to* the series itself. Thus, the sum of the series is nothing other than the series itself, also known as .20579. … By using various weird definitions one can learn to add and multiply such series with each other without having to pretend that one is really working with finitely given limits. The point is that Cantor gave up the pretense that the real numbers are primarily finite lengths. He treated them rather as arbitrary infinite series of the form *±n.r*_{1}*r*_{2}*r*_{3}*r*_{4}….

Dedekind also defined real numbers in terms of infinite sets. His approach was to characterize a real number as a *cut* [*L, R*] of the rational numbers. The idea is that every rational is either in *L* or in *R* and every member of *L* is less than every member of *R.* For instance, the square root of two would be represented by the cut [{*a/b: a*2/*b*2 < 2}, {*a/b*: *a*2/*b*2 > 2}]. The crucial thing about Dedekind’s definition of real number is that, again, the real number itself is an *infinite* set. To be more precise, a Dedekind real number is a *pair* [*L, R*] of infinite sets.

It is a curious fact in the history of mathematics that Dedekind’s definition of real numbers is taken over almost unaltered from the Eudoxian theory of proportion given in Euclid’s *Elements*, Book V. The problem Eudoxus had been concerned with was how we can compare and manipulate ratios (such as the *d:s* ratio mentioned above) that are not equal to the ratio of any two natural numbers. His solution was, essentially, to regard an irrational ratio *X:Y* as a cut of the form [{*m:n* | *mY* < *nX*}, {*m:n* | *mY* > *nX*}]. One can see that this makes sense if one realizes that *m/n < X/Y* if *mY* < *nX*, and likewise for >.

The difference between what Eudoxus and Dedekind did is that Eudoxus thought of the ratio between two magnitudes as the fundamental thing, with the description in terms of infinite sets arising only in a practical and *potentially infinite* way (since one would not, in practice, ever need *all* of the members of each side of the cut). Unless someone had constructed two specific magnitudes to be compared, the equivalent cut had no meaning … as it was an infinite, and thus unreal, thing.

Dedekind, on the other hand, accepted the *actually infinite* sets of the cut as fundamental. Whether or not one has a particular trick for constructing a length that drops a point down into the cut’s gap is immaterial. All the different actually infinite cut-sets exist in the Mindscape, and all the real numbers are already there, whether or not they can be finitely named or constructed.

The point is that the only way to get a stable mathematical representation of the notion “arbitrary real number” is to represent real numbers by actually infinite sets. There is no other way to get an absolute foundation of the real number system in terms of discrete mathematical objects.

Once it was realized that real numbers can be represented in terms of infinite sets, the dam broke. Ten years after Cantor’s death it was already a commonplace that every mathematical object can be represented by a set. If you have ever picked up a mathematics text in any field, be it analysis, algebra, or topology, you will have noticed that the book begins with a short chapter or section on set theory. This is because everything the book mentions can be best represented as a set.

For the Pythagoreans everything was a natural number. Their belief became untenable when it was realized that certain things are in their inmost essence infinite. The modern mathematical credo called Cantorism asserts that everything (at least everything mathematical) is a set.

Just as the existence of the actual infinite forces a revision of the Pythagorean position, the existence of the Absolute infinite forces a revision of the Cantorian position. If there are, indeed, Absolutes of the kind discussed in the earlier section on the Absolute infinite, then there are things that are not sets. Set theorists are still not quite certain what to do about this. But let us not worry about Absolute infinity before discussing the transfinite.

Once we realize that the irrational numbers are fundamentally infinite, in that they can be fully grounded only on a theory of infinite sets, then it is natural to start looking at infinitely large, or *transfinite*, numbers. In Cantors words, “One can without qualification say that the transfinite numbers *stand or fall* with the infinite irrationals; their inmost essence is the same, for these are definitely laid out instances or modifications of the actual infinite.”3

A last remark on Cantorism. Just as chemistry was unified and simplified when it was realized that every chemical compound is made of atoms, mathematics was dramatically unified when it was realized that every object of mathematics can be taken to be the same *kind* of thing. There are now other ways than set theory to unify mathematics, but before set theory there was no such unifying concept. Indeed, in the Renaissance, mathematicians hesitated to add *x*2 to *x*3, since the one was an area and the other a volume. Since the advent of set theory, one can correctly say that all mathematicians are exploring the same mental universe.

### TRANSFINITE NUMBERS

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In my novel *White Light*, I describe a mountain that is higher than infinity.4 This mountain, called Mount On, consists of alternating cliffs and meadows. The curious thing about it is that even after one has climbed ten cliffs, a thousand cliffs, infinitely many cliffs … there are always more cliffs. The climbers of Mount On are able to make some progress because they are able to execute a procedure called a “speedup.” By using speed-ups they are able, for instance, to zip past the first infinity of cliffs in two hours.

How is this done? The idea is to climb the first cliff in one hour, the next cliff in half an hour, the one after that in a quarter of an hour, and, in general, the *n*th cliff in hours. Since sums to 2, we see that after two hours our climbers have passed infinitely many cliffs. But there are more, many more.

In this section we will climb up through the transfinite numbers, which are usually called *ordinal numbers*, or just *ordinals.* Typically, one describes some ordinal *a* by giving an example of an ordered set *M* such that *if* one could count *M* in the correct order, *then* one would count up to *a. a* is then viewed as the *abstract order type of M*, called for short. The ordinal is gotten from the ordered set *M* by ignoring the actual appearance of the individual members of *M* and instead concentrating on the arrangement, or order, of these members.

#### FROM OMEGA TO EPSILON-ZERO

The transfinite ordinal numbers can be thought of as arising through counting. There are two principles for generating ordinal numbers: I) if you have the ordinal number *a*, then you can find a *next* ordinal, called *a* + 1; II) if you have some definite sequence of increasing ordinals *a*, then you can find a last ordinal which is greater than all the *a*’s, called lim(*a*).

We also need a first ordinal to start with, called 0. (Strictly speaking, the second principle for generating ordinals gives us 0, since zero is the first ordinal after the empty sequence.) In any case, once we have zero, the first principle can be repeatedly applied to get the ordinal numbers 0, 1, 2, …. Now, to get past the infinite sequence of finite ordinals we use principle II to get lim(*n*), usually called ω, pronounced “omega.” (Omega is also sometimes called “alef-null.”)

Omega is the last letter in the Greek alphabet (they put zeta somewhere in the middle), which may be why Cantor chose to use it as the number after all the finite numbers. The word “omega” is somewhat familiar, as it appears in the Book of Revelations, where God is quoted as saying, “I am the Alpha and the Omega,” meaning, “I am the beginning and the end.”

Now we have 0, 1, 2, … ω. Using principle I repeatedly we get the sequence 0, 1, 2, … ω, ω + 1, ω + 2, ω + 3, … To go further, we use principle II to form lim(ω + *n*), which is usually called ω + ω or ω · 2.

You might wonder why lim(ω + *n*), ω + ω, and ω · 2 should all be the same. It turns out that there is a definite way to define addition and multiplication of infinite ordinals so that everything works out. Let me briefly explain. The ordinal number *a* + *b* is obtained by counting out to *a* and then counting *b* steps further. The ordinal number *a · b* is obtained by counting up to *a, b* times in a row. That is, *a · b* is obtained by sticking together *b* copies of *a*, treating this as an ordered set *M*, and then abstracting to get the ordinal number

As long as one sticks to finite ordinals, these operations are the same as the ordinary plus and times, and they are commutative. However, once we start working with infinite ordinals, commutativity no longer holds. Thus,

1 + ω is just the same as ω, but ω + 1 is the next number after ω. Again,

ω · 2 is two omegas placed next to each other, which gives an ordinal ω + ω, but 2 · ω is omega twos placed next to each other, which makes an ordered set with ordinal number ω.

Moving on with the ordinals, by using principle I repeatedly, we now have 0, 1, 2, … ω, ω + 1, ω + 2, … ω · 2, ω · 2 + 1, ω · 2 + 2, …. It is evident that lim(ω · 2 + *n*) should be the ordinal ω · 2 + ω also known as ω · 3. Continuing in this vein, we can arrive at ω · *n* for each finite *n*, and using principle II we form lim(ω · *n*), which should be omega copies of omega, that is, ω · ω, also known as ω2.

In order to see how to pass from ω to ω2 to ω3 and finally on to ωω, it will be useful to look at the pictures in Figure 41. I will now describe four sets of points, *M*_{1}, *M*_{2}, *M*_{3}, and *M*_{ω}. In each case, the set of points can be thought of as lying between zero and one on the real number line, and each *M _{i}* will be such that if we view it as an ordered set and abstract to form the ordinal number

*M*, we arrive at ωi. One may wonder how to fit an ordered set with ordinal number ω2 in between zero and one on the real line, since the space available seems to be finite. The trick is based on Zeno’s paradox: if you start out going from zero to one, but do it by first going halfway, and then going half the remaining distance, and then half the remaining distance, and so on … always going just half of what’s left … then it will take you omega steps to get there. This is basically what the first picture shows.

_{i}The second picture is obtained by plugging a copy of the first picture into each of the spaces between points in the first picture. The third picture is obtained by plugging a copy of the second picture into each of the spaces between points in the first picture …. at least, this is what we would say if we think of ω3 as being ω2 · ω. If, on the other hand, we think of ω3 as being ω · ω2, then we would say that the third picture is obtained by plugging a copy of the first picture into each of the spaces between points in the second picture, thus obtaining ω2 copies of *ω* (as opposed to ω copies of ω2.) Both come to the same thing, since although ordinal multiplication is not commutative, it *is* associative. The fourth picture is obtained by first continuing the process started in the first three pictures endlessly … and then putting a copy of each of the pictures together.

I have written next to each picture the real number coordinates of the points involved if we think of the interval as the unit interval on the real line. These set definitions are not particularly important for us, but what *is* important is to realize that a transfinite arrangement of points can be fitted into a finite space. By using Zenonian squeezing we can sort of see an ordering of type ω2 all at once! Transfinite ordinals are not really so inconceivable after all.

It turns out that one can actually fit *any* countable ordinal into a picture of this nature. (In the next subsection we will look at the uncountable ordinals.) But the illustration of ω3 is already something of a mess, and had I not used the arrow symbolism, the picture of ωω would be so lacking in detail as to be quite uninformative. We will look at a different technique for picturing ordinals shortly.

But first let me describe just how far we want to go in this section. One way of characterizing ω2 is that it is an ordinal *a* such that ω + *a* = *a.* This can be seen by thinking of ω2 as being ω + ω + ω + ω + …. Clearly, putting an “ω+” in front of this symbol changes nothing. In point of fact, ω2 is the *first* such ordinal.

What about the first ordinal *a* such that ω · *a = a*? If we take ωω to be ω · ω · ω · ω …, we can see that placing an “ω ·” in front of ωω changes nothing. Or, assuming that the familiar laws of exponents hold for ordinals (and they do), we can reason that ω · ωω = ω1 · ωω = ω1+ω = ωω, since 1 + ω = ω. As before, it is possible to prove that ωω is the *first* ordinal *a* such that *ω · a = a.*

The first ordinal *a* such that *ωa* = *a* is called ∈_{0}, pronounced “epsilon zero.” Simply by manipulating symbols, we would expect that ∈_{0} has the following form:

Evidently, putting such a symbol in the exponent position over an omega does not change anything, since a stack of omegas 1 + ω high is the same as a stack ω high.

But we can describe epsilon zero a little better. Suppose that *ab* is defined to mean “an exponentiated stack of *a* many *b*’s.” *ab* is pronounced, “*b* tetrated to the *a.*” The name tetration is used since *tetra* is the Latin root for *four*, and tetration occurs in fourth place in the logical progression: addition, multiplication, exponentiation, tetration. You don’t ordinarily hear much about tetration because it is so powerful an operation that tetrating even very small numbers with each other produces inordinately large numbers. A tetration is worked out below.

42 = 2(2(22))

= 2(24)

= 216

= 64, 536

Note that we must be careful to associate from the top down, rather than from the bottom up, if we want to get the largest possible numbers.

As further examples, 33 is 3(33) = 327, which is just under eight trillion. 310 is 10(1010) = 10billion, which is the number that is written by putting a one and then *ten billion* zeros (as opposed to, say, a million, which is written by putting a one and then *six* zeros. Now, *2ω* is just *ωω.* And 3ω = *ω(ωω)*, which is kind of hard to get at. One way of visualizing this number is to go back to the picture of ωω, and to imagine replacing each of the dots on the line by the symbol “ω.” to get the product of *ωω* omegas.

In any case, the point of all this is that ∈_{0} is ωω. And the countable ordinals don’t stop there. If, for instance, one could make some sense of the following symbol one would have an even larger ordinal.

In trying to think of bigger and bigger ordinals, one sinks into a kind of endless morass. Any procedure you come up with for naming larger ordinals eventually peters out, and the ordinals keep on coming. Finally, your mind snaps, and maybe you get a momentary glimpse of what the Absolute infinite is all about. Then you try to formalize your glimpse, and you end up with a new system for naming ordinals … which eventually peters out….5

But this ωω is just the beginning. Let us look at a different type of picture of ωω. Suppose that we let *PN* be the set of all polynomials in *x* with natural number coefficients. Examples of members of *PN* would be *x, x* + 3, 5*x*2 + 2*x* + 4, 3*x*8 + 6*x*3 + 163. Now, suppose that given two polynomials *p*(*x*) and *q*(*x*) from *PN*, we define the following order relation: *p*(*x*) < _{bep} *q*(*x*) if and only if the graph of the polynomial *q* eventually manages to get above, and stay above, the graph of the polynomial *p.* (The letters “bep” stand for “by-end-pieces.”)

The reason I have brought all this up is that if we take *PN* with the <_{bep} ordering, then the ordinal number *PN* of this ordering is *ωω.* The correspondence is simplicity itself: the polynomial *p*(*x*) represents the transfinite number *p(ω*) (with the one stipulation that the coefficients of the polynomial must be moved to the right in the translation process that is, 3*x*2 + 2*x* is to correspond to ω2 · 3 + ω · 2).

In Figure 42 I have illustrated the fact that 0 < _{bep} 1 < _{bep}2 < _{bep}*x* + 1 < _{bep}2*x* < _{bep}*x*2 < _{bep}*x*2 + *x* < _{bep}2*x*2 < _{bep}*x*3 < _{bep}*x**x* < _{bep}3*x*, where 3*x* means *x* tetrated to the three, *x*(*x**x*).

Strictly speaking <_{bep} has only been defined for polynomials, but it is evident how we can extend it to arbitrary expressions or functions in *x.* If we allow tetration as a standard operation and let *PPN* be the set of all pseudo-polynomials formed by using natural number coefficients and tetration, as well as exponentiation, then it is not difficult to see that *PPN* is ∈_{0} when *PPN* is ordered by <_{bep}. An example of a pseudopolynomial from *PPN* would be (5*x*) + 2(3*x*)4 + *7(3x)* + (2*x*)8 + 13(2*x*)2 + 11 (2*x*) + 3*x*7 + 9*x*3 + 2*x* + 78. Note that *xx* represents ∈_{0}.

The by-end-pieces ordering was first studied by DuBois-Reymond, whose work is interestingly presented in G. H. Hardy’s book, *Orders of Infinity.6* Felix Hausdorff improved upon Reymond’s techniques in his monumental series of papers, “*Untersuchen über Ordnungstypen*,” in the early 1900s. The term “by-end-pieces ordering” is taken from Kurt Gödel, whose research revived interest in this ordering in the 1960s.7

We could really restrict our attention to functions from the natural numbers into the natural numbers. It is amusing to represent ordinal numbers as stacks of such functions, filling in the lines between graph points in the natural way. In order to make the pictures look nice, you leave out parts of the lines to avoid crossings. I call these pictures “ziggurats” because they look like Babylonian towers or Aztec pyramids. Figure 43 is a ziggurat of height ω2, and Figure 44 shows one of height ω3. In the latter I have not drawn in all the lines, so that it is not so confusing. In every case, the missing line is a line that would hug the line right underneath itself as closely as possible.

#### THE ALEFS

The famous mathematician David Hilbert used to illustrate his popular lectures with stories about a hotel with infinitely many rooms.8 This mythical hotel, usually called Hilbert’s Hotel, is supposed to have omega rooms: Room 0, Room 1, Room 2, …, Room *n*, and so on. As in the last section, it is convenient to start counting with 0.

To fix the ideas, I have drawn a picture of Hilbert’s Hotel in Figure 45. In order to fit it on the page, I have assumed that each floor is equipped with a science-fictional space condenser, a device that makes each succeeding story two-thirds as high as the one before. The shrinking field also affects the guests. Thus, although the ceilings on Floor 3 are only two or three feet high, the space condenser on that story shrinks the guests to one or two feet, and they are perfectly comfortable. I will leave it as an exercise for the reader to check that if the first floor is ten feet high, and each successive floor is two-thirds as high as the one before, then the total height of the hotel’s ω stories is thirty feet.

One of the most paradoxical things about Hilbert’s Hotel is that even after it fills up, more and more people can be squeezed in, without making anyone share a room! Say, for instance, that ω guests have arrived, and every room is occupied with a guest *n* in each Room *n.* Now, say that one more guest arrives: Guest ω. How to fit him in?

Easy! We put Guest ω in Room 0, which is emptied by moving Guest 0 to Room 1, which is emptied by moving Guest 1 to Room 2, which is emptied by …

Fine! But what if there had been an *infinite* number of new guests? Even such a procession of ω + ω guests could be lodged in Hilbert’s Hotel. We simply put the first ω guests in the even-numbered rooms, and the next ω guests in the odd-numbered rooms.

Now it is in fact possible, by suitable rearrangements, to fit ω2, ωω, or even ∈_{0} guests into Hilbert’s Hotel. But eventually we do reach a limit to this wonderful hotel’s powers of absorption: *alef-one*, also known as ℵ_{1}. Alef-one is a hard number to describe. One way of putting it is that alef-one is the first ordinal number *a* such that no possible rearrangement can fit a set of *a* guests into ω rooms. Alef-one represents an order of infinity that is *essentially greater than* ω in a way that ω + ω is not.

To get a better idea of alef-one, let’s go back to the idea of people climbing a mountain as high as all the ordinals. How hard will it be for them to get to alef-one?

We will assume that, in the marvelous land where Mount On rises, the climbers can attain any desired finite speed. Then, as we mentioned at the beginning of the “Transfinite Numbers” section, they can scale the infinity of cliffs leading up to ω in a finite time. If they keep accelerating so that the *n*th cliff only takes hours, then they reach *ω* after two hours. By repeating this they could reach *ω* + ω in four hours, or, if they did everything four times as fast, they could actually get to ω + *ω* in one hour. Turning to Figure 46, we can see how our climbers could even reach ω2 in one hour. The idea is to devote to each of the *ω2* cliffs the alloted finite interval of time. (Thus, for instance, the stretch between ω · 2 and ω · 2 + 1 is to be covered during the time interval between hour and hour.)

Now, the point I want to make is that these climbers could never reach alef-one. There is no way that we can fold together various finite bursts of speed and cover alef-one cliffs in a finite amount of time. The only way to get out to alef-one is by actually going ahead and travelling alef-one miles per hour.

One last picture of alef-one. Go back to Figure 42. In this picture we saw how various ordinals can be represented as sequences of functions ordered according to steepness (the <_{bep} ordering). How long a sequence of functions can be found with each function steeper than all the ones before? At least alef-one. That is, if *S* is a set of functions so that for every *g* (no matter how steep), there will be an *f* in *S* that is steeper, then *S* must have at least alef-one members.

It is now time to give a formal definition of alef-one. This definition hinges upon the notion of cardinality. Given two ordinals *A* and *B*, we say that *A has the same cardinality as B* if there exists a one-to-one map from *A* onto *B.* (When I speak of a map from *A* onto *B*, I really mean a map from the set of ordinals less than *A* onto the set of ordinals less than *B*.)

In our discussion of Hilbert’s Hotel we learned, for instance, that ω + ω has the same cardinality as ω. For there is a way of matching up, in a one-to-one fashion, the members of {0, 1, 2, … ω, ω + 1, ω + 2, …} with the members of {0, 1, 2, …}. That is, there is a one-to-one map from ω + ω onto ω. Now, alef-one is defined to be the first ordinal with cardinality greater than ω. Alef-one is the first ordinal that cannot be mapped one-to-one onto ω.

In general, we say that the ordinal number *A* is a *cardinal number* if and only if *A* does not have the same cardinality as any *B* less than *A*. All of the natural numbers are cardinals. There is, for instance, no way to find a *one-to-one* map from three onto two. (Keep in mind that we commonly identify the number *N* with the set {0, 1, …, *N* − 1} of *N* numbers less than *N*.)

*ω* is, of course, a cardinal number. The infinite number *ω* cannot be mapped one-to-one into any of the finite numbers before it. No finite hotel, no matter how large, can sleep infinitely many guests one to a room.

The infinite cardinals are also called *alefs.* In general, alef-*a*, or ℵ_{a}, means the *a*th infinite cardinal. Thus, alef-null, or ℵ_{0}, is just ω, the first (0th) infinite cardinal. It turns out that just as we can always find more ordinals, we can always find more cardinals. After ℵ_{0} come ℵ_{1}, ℵ_{2}, ℵ_{3}, … ℵ_{ω}, ℵ_{ω+1}, … ℵ_{ωω}, … ℵ_{ℵ1}, … ℵ_{ℵω} …, and so on. After a while we can even find a number θ such that θ = ℵ_{θ}. One way of getting such a θ is indicated below.

Is this the end? It’s never the end, when you’re discovering transfinite numbers. After *θ* come ℵ_{θ+1}, ℵ_{θ+ω}, ℵ_{θ} + ℵ_{ω}, and so on and on, world without end.

According to the Reflection Principle, it is impossible ever to conceive of an end to the ordinals. We do have a symbol Ω, the big Omega, which we use to stand for the Absolute Infinity that lies beyond all the ordinals. But Ω is inconceivable. The Reflection Principle makes this precise by saying that any description *D* of Ω that we might come up with will apply to some ordinal *a* short of Ω.

Ω is called Absolute Infinity because it is not a relative notion. The line of ordinals leading out to Ω contains *all* the ordinals, all the possible stages of counting. It is because every possible ordinal occurs before Ω that Ω is not really a definite ordinal number. Confusing talk! If you would like to read more about Ω and the transfinite numbers plan to take Excursion I.

### INFINITESIMALS AND SURREAL NUMBERS

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The best-loved of Zeno’s paradoxes is the one that states that you can never leave the room you are in. For, Zeno reasons, in order to reach the door you must first traverse half the distance there. But then you are still in the room, and to reach the door you must traverse half the remaining distance. But then….

What exactly is paradoxical here? The problem seems to be that we can analyze the passage from chair to door in two ways. First, in the normal way, as a unit, a single undivided action, as a 1. Second, as the limit of an infinite sequence of actions, as the sum of a series The modern tendency is to resolve Zeno’s paradox by insisting that the sum of the infinite series *is equal to* 1.9 One then goes on to add that since each of the successive steps is done in half as much time, the actual time taken to complete the infinite series is no different from the real time taken to leave the room.

But still … there is some residue of dissatisfaction. The feeling is that if you just keep halving the distances you never really *get* to the door. You may get arbitrarily close, but you never quite reach the limit.

The paradox can be expressed in a different way. In our ordinary real number system, we say that the number *K* with decimal expansion .99999 … is the same as 1. An informal argument for this is sketched below.

But maybe this argument is misleading. What if there is some number, call it 1 − l/ω, that is greater than any finite string .9 … 9 of nines, yet less than 1? If *K* were actually equal to 1 − l/ω, the informal argument used in the last paragraph would not work, for this argument overlooks the fact that the difference between 10*K* and 10 is ten times as great as the difference between *K* and 1. There is a residual infinitesimal quantity below that does not get canceled out.

Intuitively, nothing could be more natural than to go ahead and talk about l/ω, 1/ℵ_{1}, and so on. Just as we move from the natural numbers to the fractions and then on to the reals, should we not be able to move from the whole ordinal numbers to some richer number field?

Curiously, Cantor himself was very much opposed to this step. When a fellow mathematician attempted to use Cantor’s transfinite numbers to develop a theory of infinitely small quantities, Cantor accused him of trying to “infect mathematics with the Cholera-Bacillus of infinitesimals.”10 Cantor even constructed a proof that no number can be infinitesimal. This proof, however, is just as circular and worthless as finitist attempts to prove that no number can be infinite. In both cases, the desired conclusion is smuggled in as part of the definition of “number.”

Why was Cantor so vehemently opposed to infinitesimals? In his valuable essay, “The Metaphysics of the Calculus,” Abraham Robinson suggests that Cantor already had enough problems trying to defend transfinite numbers.11 It seems likely that, consciously or otherwise, Cantor deemed it politically wise to go along with orthodox mathematicians on the question of infinitesimals. Cantor’s stance might be compared to that of a pro-marijuana Congressional candidate who advocates harsh penalties for the sale or use of heroin. Yet, as we shall see, there is almost as much justification for infinitesimals as there is for Cantor’s transfinite ordinals.

Formally speaking, it is as consistent to say that there is a number between all of .9, .99, .999, … and 1 as it is to say that there is a number greater than all of 1, 2, 3, …. And just as we go on to find more and more ordinals piled atop one another, we can go on to find more and more infinitesimals squeezed beneath each other.

Part of the great attractiveness of Cantor’s theory stems from the fact that all of his transfinite numbers can be seen as steps towards the single Absolute Infinity that lies beyond them all. Indeed, as I have mentioned before, modern formalizations of set theory often proceed by introducing a symbol Ω for Absolute Infinity and by then assuming the Reflection Principle: every conceivable property of Ω is shared by some ordinal less than Ω. Thus, for instance, since we know that Ω is greater than all the finite numbers *n*, we know by Reflection that there must be some existing ordinal, call it ω, that is also greater than all the finite *n.*

The Reflection Principle is really a different way of saying “Ω is inconceivable.” For “Ω is inconceivable” is the same as “there is no conceivable property *P* that uniquely characterizes Ω,” and this is the same as “whenever *P* is a conceivable property of Ω, then there must be other ordinals also enjoying property *P*.” Ω can never be the *first whatever*, just as it can never be the *only whatever* … whatever *whatever* may be.

To sum up, we justify Cantor’s transfinites from the two assumptions: i) there is an Absolute Infinity, Ω; ii) Ω is inconceivable.

A skeptical reader could, quite rightly, demand to know how it is possible to discourse rationally about an inconceivable object like Ω. I would respond that Ω is a *given*, an object of our immediate pre-rational experience. And to use the tools of symbolic logic to investigate an empirically existing phenomenon is not to commit a category mistake, any more than it is a categorty mistake to look at living cells through the inanimate lenses of a microscope.

We have a primitive concept of infinity. This concept is inspired, I suspect, by the same deep substrate of mind that conditions religious thought. Set theory could even be viewed as a form of exact theology. By means of the set-theoretic analysis of Absolute Infinity, we attain knowledge of many lower infinities—the transfinite ordinals and cardinals.

I wish now to call the reader’s attention to a different sort of absolute: *Absolute Continuity.* Perfectly continuous space.

The basic intuition about an Absolutely Continuous line is that such a line cannot be conceived of as a set of points. Zeno expresses this intuition in his paradox of the arrow.12 The paradox of the arrow seems to constitute a proof that space is not made of points. For, Zeno argues, consider an arrow that flies from the bow to the target. If space is made up of points, then the flight of the arrow can be decomposed into an infinite set of frozen movements, movements where the tip of the arrow successively occupies each of the points between bow and target. The problem is that while the arrow is at any *one* fixed point, say the halfway point, the arrow is motionless. How can the flight of the arrow be a sequence of motionless stills? Where did the motion go?

A movie of an arrow’s flight is, of course, a sequence of motionless stills. But this does not disturb us, as we realize that the arrow moves in between the pictures. The problem Zeno raises is that if space is made of points, and if a still is taken at *each point*, then there is no possibility of “moving between the pictures” … because there is *nothing between the pictures.*

Zeno’s way out of the paradox is to deny that space is really made up of points. As a Parmenidean monist, Zeno viewed space as an undivided whole that cannot really be broken down into parts. We can find scattered locations in space, but space is always more than the sum of these isolated points. One can pick out higher and higher infinities of points from an Absolutely Continuous tract of space, but there will always be a residue of leftover space, of continuous little pieces, infinitesimal intervals over which the actual motion takes place.

This view of space has been held by several philosophers since Zeno, notably C. S. Peirce and, perhaps, Kurt Gödel. Gödel distinguishes between the set of points described in set theoretic analysis and the continuous line of space intuition: “According to this intuitive concept, summing up all the points, we still do not get the line; rather the points form some kind of scaffold on the line.”13

Peirce goes further than this. According to him, a truly continuous line is so richly packed with points that no conceivable set, no matter how large, can exhaust the line. There should not just be *one* point between all of and 1. There should be ω points, ℵ_{1} points, Absolutely Infinitely many!14

The early set theorist Felix Hausdorff demonstrated the logical possibility of such an Absolutely Continuous ordering. Hausdorff’s construction goes as follows. We are to imagine a superdictionary of a very special sort: i) All the words in Hausdorff’s dictionary will be spelled using only As and Bs. ii) Each word will be of Absolutely Infinite length, with one letter for each ordinal, iii) Each word must end with the letters BAAAAA …, with, that is, a single B leading off an Absolutely Infinite tail of As.15

Now, if we arrange Hausdorff’s dictionary of words in alphabetical order, we get an ordering so dense that between *any* two successive sets of words there is room for more words. You might expect AAAA … to be the first word in the dictionary, but this is not an allowable word. By rule iii, every word must have a last B before its final string of As. So is BAAA … the first word? No. ABAAA … comes first. And between these two comes, of course, ABBAAA. … In order to describe some of the longer words I will write Aω and Bω to mean strings of ω As and Bs, respectively.

AωAωBAAA … |
1/(ω + ω) |

AωBAAA … |
1/ω |

AABAAA … |
¼ |

ABAAA … |
½ |

ABBAAA … |
¾ |

AAωAAAA … |
1 − 1/ω |

BAAA … |
1 |

BBAAA … |
2 |

BωAωBAAA … |
ω/2 |

BωABAAA … |
ω − 1 |

BωBAAA … |
ω |

BωBBAAA … |
ω + 1 |

BωBωBAAA … |
ω + ω |

Exhibited above are some of the words in Hausdorff’s dictionary, along with more customary number names that are appropriate for where these words appear. These number names are expressive, but basically a bit unjustified, as we do not have in our hands a definition for how to add and multiply these words like numbers.

But quite recently, the English mathematician and puzzlist John Horton Conway has discovered an Absolutely Continuous class of numbers that do have a simply defined addition and multiplication. Conway’s new numbers are called *the class of surreal numbers*, or simply *No*.16

Conway speaks of his numbers as being “born” on an endless succession of days, one day of creation for each ordinal. In general, on the *ath* day, new numbers are placed in all gaps between successive sets of surreal numbers born on earlier days. In Conway’s ingenious system one can write down and manipulate virtually any wild number one might think of:

or what have you. He even gets a definition of the traditional symbol ∞, for potential infinity. ∞ is defined as the gap between the finitely large and the infinitely large surreal numbers, and Conway derives the weird equation, which almost magically ties together potential infinity ∞, the simplest actual infinity ω, and the Absolute Infinite Ω.

Although Conway’s surreal numbers are aesthetically pleasing and philosophically significant, they have not attained any wide usage among more practically minded mathematicians. One problem could be that it is hard to define the higher-order operations (such as exponentiation and tetration) correctly on the number field No.

Instead of using the surreal numbers, those mathematicians who need infinitesimals use a smaller, and somewhat different, extension of the reals, the so-called *hyperreal numbers.* These numbers, also known as non-standard numbers, were systematically introduced by Abraham Robinson in the 1960s.17

Conway’s numbers are introduced as “gaps” between pairs of sets. Thus, is (0, 1, 2, …|… ω/4, ω/2, ω) and, more obviously, 1/ω is Robinson’s hyperreal numbers are best thought of as sequences of reals. Thus, a hyperreal is a function *f* from the set *N*+ of positive natural numbers (*N+* = {1, 2, 3, …}) into the set *R* of real numbers.

This is convenient, since all of the operations on the ordinary reals can be carried over in a “pointwise” fashion to the functions that we call hyperreal numbers. Thus, *f* + *g* is the sequence such that *f* + *g*(*n*) = *f*(*n*) + *g*(*n*), and *fg* is defined by *fg*(*n*) = *f*(*n*)*g*(*n*).

As was already mentioned in the “From Omega to Epsilon-zero” subsection, we can order such functions by-end-pieces: *f* < _{bep} *g* means that the graph of *f* eventually gets under, and stays under, the graph of *g.*

Figure 50 is a drawing of some of the functions representing hyper-real numbers. Notice that “l/ω” is given by the function *I*(*n*) = 1/*n*. Any standard real number *r* will be represented by a constant function *c*_{r}(*n*) = *r*. It is easy to see that for any positive real *r* we will have *c*_{0} < _{bep}I < _{bep}*c _{r}*, which is to say that

*I*represents an infinitesimal.

As is well-known, Newton and Leibniz used both infinities and infinitesimals to develop the differential and integral calculus. They used the symbol “∞” for infinity, and the symbol “*dx*” for an infinitesimal number. To put a better face on things, Leibniz assured his correspondents that his ∞ was merely a potential, or “syncategorematic,” infinity. Newton called his infinitesimal *dx* a “fluxion.”

Bishop Berkeley derided Newton’s fluxions as “the ghosts of departed quantities,” but the progress of infinitesimal analysis continued unimpeded throughout the eighteenth century. No one was quite sure it was all right to use infinities, but the process gave the right answers. As Jean Le Rond d’Alembert said, “*Allez en avant, la foi vous viendra*.” Finally, in the mid-nineteenth century, Weierstrass and others found a way to avoid the use of actual infinities in calculus.18 Let us briefly describe the technique that was adopted.

In calculus one discusses different sorts of calculation processes *C*(, ,) whose outcome depends on what numbers are fed in. A typical use of infinitely large and infinitely small numbers is to say that for certain real numbers *a* and *b*, the outcome of *C*(∞, *dx, a)* is *b*. Both the derivative and the integral are defined in this way: as calculations that accept a real number (or numbers) and some infinitely large and/or small numbers as input, and that then give some definite real number as output.

Weierstrass’s technique was to replace a statement such as “*C*(∞, *dx, a*) = *b*” by “if we take *I* to be a *large enough* real number, and *i* to be a *small enough* real number, then the value of *C(I, i, a)* will be *very close* to *b.*” By letting *I* grow and *i* shrink, one is supposed to be able to bring *C* (*I, i, a)* to within any desired degree of closeness to *b*. This is called the *limit process.*

Of course, if one believes in infinitely large and infinitely small numbers, it is much simpler just to ask if *C*(∞, *dx, a)* equals *b.* But so great is the average person’s fear of infinity that to this day calculus all over the world is being taught as a study of *limit processes* instead of what it really is: *infinitesimal analysis.*

As someone who has spent a good portion of his adult life teaching calculus courses for a living, I can tell you how weary one gets of trying to explain the complex and fiddling theory of limits to wave after wave of uncomprehending freshmen.

I often think of C. H. Hinton’s words from a similar context:

“How pleasant it would be to let pass away some of that verbiage I learnt at school—learnt because teachers must live, I suppose. The apeing and prolonged caw called grammar, the cackling of the human hen over the egg of language—I should like to unlearn grammar.”19

But there is hope for a brighter future. Robinson’s investigations of the hyperreal numbers have put infinitesimals on a logically unimpeachable basis, and here and there calculus texts based on infinitesimals have appeared.20

### HIGHER PHYSICAL INFINITIES

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Suppose that our universe were, in fact, more than ω miles in diameter. Could you ever travel a transfinite number of miles from Earth?

This *would be possible*, assuming you had the ability to accelerate up to any speed short of the speed of light (some .7 billion miles per hour). The technique hinges on Einstein’s Special Theory of Relativity. As you travel closer and closer to the speed of light, your own time runs slower and slower relative to the rest of the universe. By a suitable acceleration, you could travel, say, ω billion miles in what feels like four hours.

The first billion miles (done at about .4 the speed of light) would seem to take two hours. The next billion miles (done at about .6 the speed of light) would take only an hour of your time. The third billion (done at .7 the speed of light) would take you only half an hour. And so on. After four hours you would be past milestone ω.

In my novel, *White Light*, I describe such a journey from the viewpoint of a massless soul and spirit who is able to accelerate himself endlessly by some obscure force of will. Let me reprint the description of his journey, purportedly made with a fellow spirit named Kathy:

The first part of the trip was dull. Although we were accelerating steadily, it still took an hour to get out of the solar system. And then we had an hour and a half of vacuum till the next star.

About three hours into our trip it began to get interesting. Objectively we were doing about .7 the speed of light. Because of our distorted time and length standards, it felt like we were doing three times that. Weird relativistic effects began setting in.

It seemed like we were looking out of a cave. All behind us and on both sides of us there was the dead absolute nothing called ‘Elsewhere’ in relativity theory. The stars had somehow all scooted their images around to in front of us. We accelerated harder.

The thousand light-year trip across the galaxy only seemed to take half an hour. But what a half-hour. I would be looking out our speed-cone at the vast disk of stars that lay ahead of me … most of them clinging to the edge. Slowly one of the stars would detach itself from the clustered edge and accelerate along a hyperbolic path towards the center, then, ZOW, it would whip past us and go arcing back out to the edge of our visual field.

There was a pattern to the flicker of passing stars, and I began to get into it. It was like listening to the clicking of train wheels. Everything but the swooping pulses of light faded from my attention. I pushed to make the flickering come faster.

There were patterns to the flicker … star clusters … and as we accelerated more I began to see second and third order patterns. Suddenly the stars stopped. We were out of the galaxy.

Our visual field had contracted so much that I felt myself to be looking out of a porthole. There was dark on all sides and I knew fear. My back was a knot of pain, but I drove myself to accelerate more and more, to make the porthole smaller.

A few squashed disks of light tumbled out from infinity and whizzed back. Then more and more came twisting past. Galaxies. I felt like a gnat in a snowstorm. We flew through some of the galaxies. Inside was a happy blur. We were going much too fast to see the individual stars hurtle past.

We pushed harder, harder. We hit a galaxy every few seconds now, and as before I began to detect higher-order patterns in the stroboscopic flicker.

From then on that was all I could see … a flicker which would build and build to an almost constant flash, abruptly drop in frequency, and then build again. At the end of each cycle we reached a higher level of clustering and the light became brighter.

I was on the ragged edge of exhaustion. The strobing was building castled landscapes in my mind. My lucidity was fading fast as I stared into the more and more involuted blur of light before me. I tried to make it come faster.

There was still a certain depth to the pattern of light ahead of us, but I noticed that the harder I pushed the acceleration, the shallower and more two-dimensional the scene in front of me became. I concentrated on flattening it out.

The energy to push no longer seemed to emanate from me or from Kathy. It was as if I were somehow ram-jetting the incoming light right through us … applying only a certain shift of perspective to move us even faster.

With a final effort we turned the universe into a single blinding point of light.21

I have reprinted such a long excerpt so as to make the idea of travelling out past infinity feel a bit more natural. Of course, actually to do such a trip in a real, and massive, spaceship would require an infinite amount of energy. But this could perhaps be taken care of by scooping in star after passing star for fuel!

A side effect of using relativistic time dilation to travel ω miles is that one would finish the trip not only ω miles away from earth, but also ω years in the future. The concept of a transfinite cosmic time scale has appeared in some modern astrophysical discussions. In particular, it has been predicted that if one were to fly at a rotating black hole, one would bounce off and land ω years in the future—in, effectively, a different universe.22

A different method of achieving very high physical infinities in the large is to postulate the existence of many parallel universes. Indeed, if our universe is determined by ℵ_{0} bits of information, and if every possible universe exists, then there must be at least ℵ_{1} other universes. Hugh Everett’s notorious Many-Worlds Interpretation for quantum mechanics postulates a situation like this. One difficulty for the doubting Kantian is that it is, in principle, impossible to detect the other parallel worlds.

The idea of one universe that lasts ℵ_{1} years is about the same as the idea of ℵ_{1} universes that last ω years each. In either case, one has to do something very extraordinary to break out of the particular space-time one starts in. Whether one calls this extraordinary action “travelling infinitely far into the future” or “jumping to a parallel time-stream” is, perhaps, only a matter of taste.23

What about higher infinities in the small? As is discussed in Excursion I, Cantor proved in 1873 that mathematical space contains at least ℵ_{1} points. So, if our physical space is actually rich enough so that every real number (such as 3.14159 …) represents a distinct point in space, then there are at least ℵ_{1} points in space. Whether mathematical space might contain more than ℵ_{1} points is still an open problem: the Continuum Problem.

Assuming that our physical space is, in fact, the same as mathematical space, Cantor was led to a very strange speculation: that matter-objects and aether-objects are made up of, respectively, ℵ_{0} and ℵ_{1} “atoms” each. Let us see how he arrived at this position.24

Cantor believed that since any tiny ball-type atom can be cut in half, the ultimate constituents of matter must be point-sized objects, which he called *mass-monads.* (The word “monad” is taken from Leibniz, who in his *Monadology* describes the universe as a collection of simple and indivisible monads acting in harmony.)25 Cantor held that any piece of matter is made up of ℵ_{0} mass-monads, packed densely together like the rational numbers on the number line.

In Cantor’s time it was widely believed that there is another fundamental substance besides matter: *aether.* Cantor did not regard the aether as an all-pervading substance filling all of space. He believed, rather, that the aether is clumpy, and that such phenomena as light, heat, electricity, and magnetism could be explained in terms of tendrils and globs of aether.

In so far as aether should be a finer and subtler substance than matter, Cantor proposed that each chunk of aether is made up of ℵ_{1} *aether-monads.* Thus, mass is thought of as something similar to a heap of sand, a congeries of bright dots, while aether is like water, a continuous substance that can trickle through the infinitesimal interstices in matter.

Can anything be made of Cantor’s theory? Present-day science has no need of aether, or of any other higher types of matter. But something may yet come of Cantor’s theory. Peirce, for instance, has suggested that the mind-body problem might be solved by regarding the human mind as being able to act on Cantorian aether-objects. Perhaps we have ectoplasmic souls, each made of ℵ_{1} aether-monads!

Infinities much larger than ℵ_{1} could occur in the small, if space were to be Absolutely Continuous in the sense discussed in the previous section. It is hard to imagine ever observing such nested infinitesimals, but some useful theory, with observable consequences, may find its theoretical foundations in Absolutely Continuous space. Or it could be that particles can be viewed as singularities of varying transfinite order.

### PUZZLES AND PARODOXES

(*Answers on page 297*)

1. Describe how you would organize your time so as to travel ω + ω miles in one hour.

2. Let *a* be some real number between − 1 and 1. Consider the sum *S* of the geometric series given by the equation *S* = 1 + *a* + *a*2 + *a*3 + … Prove that *S* is 1/(1 − *a*) by multiplying both sides of the defining equation by *a*, and by then subtracting the new equation from the old. Test the formula out on and .1. What happens for *a* = 1, − 1, or 2?

3. How exactly would you fit ω · ω guests into the ω rooms of Hilbert’s Hotel?

4. Consider the guest register at Hilbert’s Hotel. Each page holds only finitely many names, and new arrivals must always sign in at the next available space. How many pages must the register have so that as long as guests can still be squeezed into the hotel (possibly by rearranging them), there is still room for the signatures of the new arrivals (without rearranging)?

5. In his short story, “The Book of Sand,” Jorge-Luis Borges describes an infinite book that i) has no first or last page.26 The book is written in a foreign alphabet, but includes illustrations on some of the pages. The narrator in the story observes that ii) each page of the Book of Sand has a different, apparently random, natural number in the corner. He also reports that iii) “The small illustrations, I verified, came two thousand pages apart.” What type of ordering would the pages of this book resemble?

6. What is the difference between ℵ_{1} · ω and ω · ℵ_{1}?

7. We say the ordinal number *a* is even, if for some ordinal *b*, 2 · *b* = *a.* Is ω even? Is ω + 4 even?

8. The ordinal number *a* is said to be *regular*, if *a* cannot be written as the sum of less than *a* numbers, all of which are less than *a* · ω is regular since it is not the sum of finitely many finite numbers, ω + ω is not regular since it is the sum of two (which is less than ω + ω) numbers (ω and ω) less than ω + ω. Ten is not regular since it is the sum of less than ten numbers less than ten, (say 3 + 5 + 2, or 9 + 1). Is ℵ_{1} regular? Only three of the finite natural numbers are regular. Which are these numbers?

9. Points in the usual Cartesian plane are given as pairs (*x, y*) of real numbers. Let the points in the *Dehn plane* be given as pairs (*x, y*) of finitely large surreal numbers.27 (Infinitely small numbers allowed, but infinitely large numbers excluded.) Show that Euclid’s Fifth Postulate fails in the Dehn plane-that is, show that there are *many* straight lines through (0, 1) that do not intersect the *x*-axis at any finite point- and conclude that in the Dehn plane a line can have *more* than one “parallel” through a given point.

10. A recent science-fiction novel states that the following is a law of nature: “Every string which has one end also has another end.”28 Is this necessarily true?

## CHAPTER THREE

## THE UNNAMEABLE

This chapter discusses three famous logical paradoxes: the Berry paradox, Richard’s paradox, and the paradox of the Liar. It is my contention that each of these paradoxes points to the existence of mental concepts that defy any exact formalization. Insofar as the human mind understands these paradoxes, it is in some sense infinite.

The first section deals with the Berry paradox—a paradox having to do with the impossibility of ever explaining exactly how we use language. This paradox has certain implications about the limitations of man-made “thinking machines.”

The second section discusses a number of topics relating to Richard’s paradox. In particular, we ask if there is, in heaven or on Earth, anything truly random—random in the sense of having no finite description, random in the sense of being infinitely complex.

The “What is Truth?” section delves into the rich variety of problems coming out of the simple sentence: THIS SENTENCE IS NOT TRUE.

### THE BERRY PARADOX1

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What is the biggest natural number you can think of? If your mind is infinite (as it may very well be), then you might insist that you can think of *all* the natural numbers, so I’d better change the question to this: What is the biggest natural number that you can describe? On the one hand, it seems that there can be no greatest number *G* that you can describe … for if you can talk about *G*, then why shouldn’t you be able to talk about *G* + 1, *G* + *G, G · G*, and *GG*? On the other hand, it seems that there must be some limit to the sizes of the natural numbers that you can describe … for whether or not your mind is really infinite, you certainly will not specifically mention each of the natural numbers in your lifetime.

This last fact seems at first to provide a way out of the dilemma posed by the question of what is the largest number I personally can describe. The fact is that I might once discuss some number *G*, but then die before ever getting around to mentioning *G* + 1, so it is not, after all, true that if one can talk about *G*, then one can invariably talk about *G +* 1 as well. But what about talk like this: “Let *u _{0}* be the smallest natural number that I will never name,” or “Let

*W*be the smallest natural number that is greater than every number that I will ever name.” Can it be that these phrases are not really names? But who can deny that the numbers

*u*and

_{0}*W*actually exist, even though their precise values will not be clear until I die?

There is a realm of natural numbers that can be humanly described, and beyond a somewhat variable transition point, there lies a whole other realm of natural numbers that cannot be singled out by any description that is short enough to be humanly comprehensible. It is these unnameable natural numbers that are of interest here.

There is, of course, a sense in which any natural number has a name—the base ten positional representation we are so familiar with. But for you or me, something like 543 … 784 cannot really serve as a description of a number if the “ … ” stands for a random string of digits stretching from here to the other side of the galaxy.

To make our original question less subject to individual variations, we will ask it about a general person rather than about you specifically; to make the question more precise, we will require our descriptions to be less than one billion words long. Thus, the question now becomes: What is the biggest natural number that a person can be told of in less than a billion words? Actually, it is more interesting to phrase the question this way: What is the first natural number that cannot be described to a person in under one billion words?

The reason I pick the limit of a billion words here is that I am about thirty-five, and I estimate that so far I have read some 300 million words and have heard some 200 million words, making a total of half a billion words. So a reasonably generous estimate of the number of words a person might absorb in a lifetime seems to be one billion. The point of my question is: What is the smallest natural number that cannot be described to a person in words?

Perhaps you have noticed that there is something fishy here. Assume there are indeed numbers that cannot be described to a person in words in the space of a lifetime; and assume that there is indeed a *least* such number, which we may as well call *u _{0}.* Now, it looks as if

*I have just described a particular natural number called u*. But

_{0}*u*

_{0}is supposed to be the first number that cannot be described in words.Paradoxes are compact energy sources, talismans. Contemplating one of Zeno’s paradoxes is like running your hand over some callipygous marble Aphrodite, or staring into a mandala. A good paradox can never be finally disposed of. The paradox touched upon in the paragraph above is a version of the problem of how we can talk about things that we cannot talk about. It was cast in the present form by an obscure librarian named G. G. Berry, who told it to Bertrand Russell, who put it this way: “ ‘The least integer not nameable in fewer than nineteen syllables’ is itself a name consisting of eighteen syllables; hence the least integer not nameable in fewer than nineteen syllables can be named in eighteen syllables, which is a contradiction.”2 The very existence of a paradox such as this can be used to derive some interesting facts about the relationship between the mind and the universe. No one has made such a derivation as boldly as Borges:

“We (the undivided divinity operating within us) have dreamt the world. We have dreamt it as firm, mysterious, visible, ubiquitous in space and durable in time; but in its architecture we have allowed tenuous and external crevices of unreason which tell us it is false.”3

Actually, since the paradoxes inhere in the very nature of rational thought, I don’t think that “we” *could* have chosen to dream a world free of paradoxes. Rather than saying that the paradoxes indicate that the rational world is “false,” I would say that they indicate that it is *incomplete*—that there is more to reality than meets the eye.

In order to reach a richer appreciation of what is involved in the paradox of the first unnameable number, it is necessary to think a bit about how one goes about naming natural numbers.

#### NAMING NUMBERS

Imagine. Pursued by a tiny girl in a yellow dress, white ducks move deliberately across a green sward sloping down to stagnant brown water. How many ducks are there?

It is surprisingly difficult to freeze and count the items in a mental image. As soon as you muster the determination and mental clarity to focus on one part of the flock of ducks, the rest of them become hazy and variable.

In Rene Daumal’s *Mount Analogue*, the character Father Sogol (who is modelled on G. I. Gurdjieff) claims that we cannot really carry more than four things in the mind at once, and gives the following example:

“1. I get dressed to go out; 2. I go out to catch a train; 3. I catch the train to go to work; 4.1 go to work to earn a living …; now try to add a fifth step, and I am sure that at least one of the first three will vanish from your mind.”4

The word “number” can be construed as meaning “that which makes numb.” The point is that the assigning of numbers to things in the world replaces vibrant confusion with hard and immutable fact. For an idealist, the surprising thing is that it is possible even to *begin* to thus “numb” the world. That is, if one believes that the whole world is a dream, an illusion, an image in some Mind … if one believes this, then it is hard to account for the identity between the numbers that different people extract from the world. If I go into the woods and count the branches on a particular dead oak that looks like a dinosaur, and if you do the same tomorrow, then our numbers will certainly agree. You can leave your home for fourteen years, and when you go back, your house is *still* the seventh on the right. The only way to account for the numerical identities among the worlds you and I dream up must be that in some sense you and I are really the same person.

But whether or not we are in some sense the same person, I still have to write the rest of this section, so I can read it and find out about the Berry paradox! The point I was just making is that we get some sort of hold on the world by assigning numbers to things. Now, the way in which we assign numbers to finite sets of things is by *counting.* In order to learn to count we memorize, during the first few years of life, a certain sequence of sounds. To count a small collection you say the successive number sounds as you point once at each member of the collection. The last sound you make is used as the name of the number, or the size or cardinality of the collection.

Any well-remembered sequence could be used as a number list. The Greeks used the letters of their alphabet. If you liked you could use the names in your phone book or the verses of the Bible. In his story “Funes the Memorious,” Borges describes a youth with a perfect memory who invents a unique name for each of the first 24,000 numbers:

In place of seven thousand thirteen, he would say (for example) *Maximo Perez;* in place of seven thousand fourteen, *The Railroad;* other numbers were *Luis Melian Lafinur, Olimar, sulpher, the reins, the whale, the gas, the caldron, Napoleon, Agustin de Vedia.* In place of five hundred, he would say *nine.* … I tried to explain to him that this rhapsody of incoherent terms was precisely the opposite of a system of numbers. I told him that saying 365 meant saying three hundreds, six tens, five ones, an analysis which is not found in the ‘numbers’ *The Negro Timoteo* or *meat blanket.* Funes did not understand me or refused to understad me.5

As Borges points out, the drawback with unusual counting sequences is not only that they are hard to learn, but also that they are not based on a system which produces an unending supply of new names.

The simplest system for naming numbers is the tally system. This system names the number *n* by a sequence of *n* strokes. Thus, the name of the number we normally call five would be /////, or stroke stroke stroke stroke stroke. The first number not nameable by the tally system in less than one billion words is just stroke stroke stroke … stroke stroke stroke, where I have left out 999,999,994 strokes so that there will be room for something else in my life.

Our familiar number system is based on memorized names for the numbers / through /////////; an ingenious system based on powers of ten is used for larger numbers. There are, in principle, definite names for every number. In the United States the name of the number we write as a 1 followed by 3 (*n* +1) 0’s is called a (Latin for *n*)-illion. Thus, if we have a 1 followed by 12 0’s, then we have a trillion, since 12 = 3(3 + 1) and Latin for three is tri. The number consisting of a one followed by 100 0’s is frequently called a *googol*, but it could also be called ten duotrentillion, since 100 = 1 + 3(32 + 1) and Latin for thirty-two is something like duo-trentum. Actually, one hardly ever uses the higher “-illion” names, and numbers which are more than thirty digits long are normally read just by listing their digits with the understanding that these digits are to be interpreted in terms of the powers of ten system.

For large numbers it is convenient to use exponential notation. Here googol is written as 10100, and one can readily pass on to *googolplex*, defined as 10googol or l0(10100). Notice that googolplex is *not* nameable in less than a billion words if we only use the ordinary “-illion” number notation for it. Clearly there will be some numbers near googolplex which are so irregular that they cannot be named in any way much shorter than reading off their digits. These numbers are really unnameable for a human being, since a number that was googol digits long would, if written out on sheets of paper, easily fill space out to the most distant visible star—since I estimate that if we packed ten billion cubic light years with books containing digit after digit, then there would only be room for some 1062 digits in these books.

Making estimates of such ridiculously large numbers is an ancient and honorable pastime. There is a well-known treatise by Archimedes called “The Sand Reckoner,” where he estimates that it would take under 1063 grains of sand to fill a sphere with a radius equal to the distance from the Earth to the Sun.6 The really interesting thing about his essay is that the Greeks did not have the notion of exponentiation. All they had was the notion of multiplying two numbers, and the largest number that they had a name for was a *myriad* (= 10,000 = 104).

So how did Archimedes manage to get out to 1063? He took myriad-myriad as his basic building block *M* (= 108). What he did then was, without using exponents directly, to find names for all of the numbers out to *M(M2)* (= 10(8 · 1016)). His trick was, for each *j* and *k* less than or equal to *M*, to introduce numbers of the *k*th order of the *j*th period. If we call the greatest number of the *k*th order of the *j*th period *A(k, j);* then Archimedes’ construction can be summarized by four rules: i) *A*(1, 1) = *M*; ii) *A*(1, *j* + 1) = *M · A*(*k, j*); iii) *A*(*k* + 1, 1) = *A*(*k, M*); iv) *A*(*k* + 1, *j* +1) *= M · A(k* + 1, *j*). Actually, Archimedes could have done better. He should have used a stronger rule than iv)—call it iv*) *A(k* + 1, *j* + 1) = *A(k* + 1, 1) · *A(k* + 1, *j*). The largest number he reached was *A*(*M, M*), which he called “myriad-myriad units of the myriad-myriad-th order of the myriad-myriad-th period,” and which is *(MM)M* = *M*(*M*2), the product of *M*2 copies of *M*. If Archimedes had used iv*) instead of iv), he would have gotten *A*(M, M*) = *M*(*MM*), the product of *M**M* copies of *M*.

One wonders whether there is any limit to the numbers that could be described on the basis of *M*, multiplication, and nested iterations of length *M*. There is a sort of double iteration process that defines what is called the *Ackermann generalized exponential G*(*n, k, j*) as follows i) *G(1, k, j)* = *j · k;* ii) *G(n* + 1, 1, *j*) = *j*; iii) *G(n* + 1, *k* + 1, *j*)= *G(n, G(n* + 1, *k, j), j).* This looks harmless enough, but it turns out that *G(2, k, j)* is the number gotten by multiplying *k* many *j*’s (also known as *jk); G*(3, *k, j)* is the number gotten by exponentiating a stack of *k* many *j’s* (also known as *kj* or *j* tetrated to the *k); G*(4, *k, j)* is the number gotten by tetrating a stack of *k* many *j*’s (sometimes known as *j* pentated to the *k*); and so on. Clearly *G(M, M, M*) is going to be a fabulously large natural number.

There is a certain formal criterion that regards the definition of *A* as singly nested and the definition of *G* as doubly nested. It turns out that the function taking *x* into *G(x, x, x)* is by-end-pieces greater than any function that can be defined using single nesting (recall the by-end-pieces ordering from “Transfinite Numbers.”) By nesting our definitions more than two times, we can get more rapidly growing functions and names for larger numbers. It would seem that the limit of what can be done here might be a number *P* that is greater than any *H*(*M*, …, *M*), where *H* is a function of *M* arguments defined by at most *M* nestings. The idea would be that one cannot systematically reach beyond *P* without using a systematic procedure that in some dimension is bigger than *M*.

This does not, however, rule out the possibility of finding a short, but non-systematic, description of some number much greater than *P*. In 1742, Christian Goldbach made the conjecture that every even number greater than two is the sum of two primes. It *is* not known whether or not Goldbach’s conjecture is true. It *is* true for all the even numbers that anyone has ever looked at: 4 = 2 + 2 (recall that 2 is the first prime number, as it has no divisors other than 1 and itself), 6 = 3 + 3, 8 = 3 + 5, 20 = 7 + 13, 84 = 23 + 61, and so on. But maybe Goldbach’s conjecture fails for some fantastically large even number. In this case “the least even number after 2 that is not the sum of two primes” could be the name of some number much greater than *P*.

In itself, the string of symbols *G(M, M, M*) is not really a name. A name should be self-sufficient, containing definitions of all the symbols and words that it uses. Of course, these definitions involve more words, but the assumption is that ultimately any description should be reducible to extremely simple statements about, let us say, writing down strokes. It is not hard to imagine how one would complete the name *G(M, M, M)* by adding a description of *M*, of multiplication, of the Ackerman generalized exponential, and so on. We denote this expanded name by [*G(M, M, M*)]. The idea is that [*G(M, M, M*)] might be something like the last few pages, along with a few more pages of further amplification. Given this full description, anyone would be able to figure out what number is being named, to the point of being able to come up with a list of *G(M, M, M*) strokes, if there were no limitation on time.

Names like [the googolth prime number] or *[G(M, M, M*)] are what we might call constructive names. There is another sort of name, [the least even number greater than 2 that is not the sum of two primes], [the millionth perfect number], or [the first *n* such that there is a string of 20 sevens that ends at the *n*th place of the decimal expansion of π]. It is not presently known whether any of these names actually names a number, since we don’t know if there are any even numbers greater than 2 that are not the sum of two primes, or if there are actually a million perfect numbers (recall that a number is perfect if it is the sum of its divisors other than itself), or if there is a string of 20 sevens anywhere in the decimal expansion of π. Trying to find the number named by each of these names involves searching through all the numbers until the right sort of number is found. But in each case, the search could be fruitless. If, however, you didn’t know this in advance, you would simply continue searching forever.

#### UNDERSTANDING NAMES

Now that we have discussed various sorts of names of numbers, let’s go back to the paradox that started all this. *u*_{0} is supposed to be the first number that cannot be described in under one billion words. It seems that the only way to avoid paradox is to assume that *u*_{0} is not the name or description of any number. This could be because i) every natural number *can* be described in under one billion words, or because ii) there is no way to extend our description of *u*_{0} to a really complete description [*u*_{0}] that can be understood by everyone.

Possibility i) is a bit odd. The idea here is that even though there are only a finite number of names that have a length under one billion words, there are infinitely many ways of *taking* these names. Given the open-ended nature of mind and language, a single name can be interpreted in many ways. This is actually less reasonable than it sounds. What possibility i) boils down to is that if *u*_{0} is a name of *n*, then it must be the name of some *m > n* as well, so that *u*_{0} itself is actually the name of infinitely many different numbers. The justification for this would be that each of the infinitely many times you say, “The first number not nameable in under one billion words,” you will mean something a little more comprehensive by *nameable.* In other words, you start out saying, “*u*_{0}” and then you say, “but that wasn’t the *real u*_{0} since I laid that name ‘*u*_{0}’ on it … what I am thinking of now is the *real u _{0}*,” and you get a bigger number; and then you keep doing the same shift over and over forever. The point is that if you think of the concept of

*naming*in endlessly more sophisticated ways, then the single name

*u*

_{0}can actually serve to name each of the natural numbers!

This way of weaseling out the Berry paradox is sometimes explicitly ruled out by requiring that the names for numbers be interpreted in one and only one definite way. In this case, we are forced to accept possibility ii). Possibility ii) says that there is no way to explicate in under one billion words what we mean by “nameable in under one billion words.” Where exactly does the difficulty lie? The problem is not in coming up with a list of all the possible combinations of under one billion words. In principle, that can be done quite mechanically. A sufficiently large machine could mindlessly print out all of the possible combinations of under one billion words without any difficulty at all. Assuming that we restrict ourselves to the million or so words that make up the English language, then there will be something like 10(6 billion) sequences of under one billion words.

Let me repeat that the problem in explicating the phrase “nameable in under one billion words” is *not* the problem of actually producing each of the 10(6 billion) possible combinations of words that might constitute a person’s lifetime word intake. The problem is, rather, this: there is no way to describe in (under a billion) words a general procedure that will translate any string of (under a billion) words into the number, if any, named by that string of words. Put differently, *there is no way for a person to describe exhaustively how he goes about transforming words into thoughts.*

For suppose that you could come up with some final description of how it is that you turn words into thoughts, names into numbers. This description, call it TRANS, would be so precise that given TRANS and any sequence of words one could, without having to think at all, apply TRANS to the sequence of words and come up with the number, if any, described by the sequence in question. Now, consider this description of *u*_{0}: [Mechanically generate one after another of the possible strings of under one billion English words. Apply TRANS to each string in turn, making a list of the numbers obtained in this way. *u*_{0} is the first number not in this list.].

This bracketed description, let us call it [*u*_{0}-TRANS], has approximately the same length as TRANS; so if TRANS could be described in under one billion words, then [*u*_{0}-TRANS] would also weigh in at under one billion words, leading to the unacceptable conclusion that *u*_{0} can be described in under one billion words. So we must conclude that any TRANS that is less than one billion words cannot code up an exhaustive or programmatic description of how to understand every English phrase shorter than one billion words.7

The bound of one billion is perhaps too big to be meaningful. Let’s drop our word limit down to 200,000, the length of a good-sized book. What we have discovered is that there can be no book-length program that will enable a computer to understand every book. One could, of course, write a “crib sheet” program that provided explicit answers to every question that could ever be asked about every book—but this program would overflow the bounds of the galaxy. The interesting fact that we have learned from the Berry paradox is that there can never be a tricky sort of program that will give a *reasonably short* description of how to understand language.

It is perhaps not too much a caricature of the later Wittgenstein’s “ordinary language” philosophy to say he argued that when people talk to each other, they are really just playing a game with noises. You make such a noise and I do this and that. But if it is *even in principle* impossible for anyone ever to formulate a complete set of rules describing how he uses language, then how can anyone assert that learning to use language is just a process of learning to play a certain game according to certain rules? To express this thought in the style of Wittgenstein: If this is just a game following rules, then why can’t anyone tell me what the rules are?

Wittgenstein’s earlier view of language seems more tenable.8 According to this “picture theory of language,” there are certain relationships among concepts and objects that we perceive in the physical and mental universe around us. In order to point these features out to each other, we use words to set up language structures that somehow mirror or model them. I think such a view of language (which is dependent on an external, objective, but undefinable, concept of “truth”) seems much more tenable than any view of language as the working out of a formalizable logical system.

It is worthwhile to give a more precise description of what the Berry paradox tells us about digital computers. The information-theoretic interpretation of the Berry paradox that I am about to describe is due to Gregory Chaitin.9 Consider a fixed machine *M.* For the sake of definiteness, let us say that *M* is a large digital computer with an interactive APL terminal, which has the form of an IBM typewriter with a thick cable leading out the back. We can think of any string of symbols that is typed on the terminal keyboard as being a program *P.* We say that *P* is an *M*-name for the number *n* if typing the program *P* causes the machine to type out the number *n* in the customary base ten notation and then stop.

The length of a program *P* is defined to be the number of key strokes necessary to input *P*, and we define the *information-theoretic complexity of n on M* to be the number *I _{M}*(

*n*) equal to the length of the shortest program that is an

*M*-name for

*n*. In other words,

*I*(

_{M}*n*) is the smallest number

*k*such that

*k*properly chosen hits on the keyboard will cause to print out the number

*n*in base ten notation.

It is not hard to see that *I*_{M}(*n*) will never be longer than the actual number of digits in the number *n*. For if we simply type the number *n* on our terminal, then we will not have told *M* to *do* anything, so *M* will be stopped, and the number *n* will be sitting there typed out on the paper in front of us. If one types the “program” “25252525252525252525252525252525,” then one is immediately in the situation of having 25252525252525252525252525252525 typed out and *M* stopped. However, we could achieve the same end result more economically by typing the instruction: “Print ‘25’ sixteen times and stop.” The latter instruction is a bit more economical since it uses 27 keystrokes instead of the 32 keystrokes of the original program. For a more extreme example, consider the case of 101000. To type out this number in base ten notation requires 1001 keystrokes. But, assuming our *M* uses the symbol “*” to stand for exponentiation, we can simply type “evaluate 10*1000” on the terminal, which will then busily type out all 1001 digits of l01000 and stop.

We now want to consider the program *R*, which has a form something like, “Print the first number that has no *M*-name as short as this program.” Now, we do not think of *M* as automatically endowed with self-knowledge, so the phrase “*M*-name” must be explicitly definied. This can be done by giving a list of the rules by which *M* actually operates—in effect, one gives the computer instructions that enable it internally to simulate its own behavior. Having done this, we can find some simply described number *r* (such as 10*10) such that the program *R* will have a length much less than *r*, where *R* is, “Simulate a machine *M* that operates as follows: [Description of *M* goes here]. Now print out the first number that has no *M*-name shorter than *r*.”

Now, since this program has length much less than *r*, it must fail to be an *M*-name for any number, otherwise a contradiction would arise. But how can a definite program such as *R* fail to name a number? Anyone who has ever tried to program a computer is familiar with the phenomena of *looping* and *endless search.* Certain programs cause the machine to go into an endless loop, never outputting anything; other programs will cause the machine to output an endless string of numbers. When a machine enters either one of these situations, then it will run forever unless some external agent turns it off.

#1: Go to #2 | If n = n, then print “n.” |

#2: Go to #1 | If n ≠ n, then stop. |

LOOPING | ENDLESS SEARCH |

If program *R* is fed into *M*, then *M* will run forever and never come up with an output. Why is this? When *M* tries to execute the program *R, M* must go through a process like this: i) list all possible programs of length *r*; ii) for each such program *P*, simulate the action of *M* on *P*; iii) if *M* acting on *P* outputs *n*, then place *n* in the set *S;* iv) let *u*_{0}(*M*) be the first number not in the set *S.* We know *a priori* that this process can never terminate, so there must be some sort of loop or endless search hidden in it. Where? The loop is in the translation step ii). For one of the programs of length less than *r* is going to be the program *R*, and when *M* tries to simulate the action of *M* on *R*, then *M* must simulate *M* trying to work through steps i) − iv) of *R …* which eventually leads to simulating the simulation of program *R* … which leads to the simulation of the simulation of the simulation of program *R* …

It is almost as if the poor machine is trying to break through to full self-consciousness with this endless regress of self-simulation. Another way of looking at this problem is to say that *M* cannot prove that there are any numbers *n* with complexity *I*_{M}(*n*) > *r*. Now, *r* is approximately equal to the complexity of a description of *M*, so one might say that *M* cannot prove the existence of any numbers with information-theoretic complexity much greater than that of *M*.

I used *u*_{0}(*M*) above to stand for the first number with complexity greater than *r.* Suppose that we use *W*(*M*) to stand for the first number greater than every number with complexity less than *r*. That is, *W*(*M*) is greater than any number that *M* can output on the basis of less than *r* strokes, where *r*, again, is a number chosen to be greater than the complexity of *M*. If we think of the output of *M* as being the number of seconds *t* that it runs before stopping, instead of being the number *n* that it prints, then we have an interesting situation. Assuming that *M* is as complex a thing as can be conceived of, then any instruction fed into *M* will be less complex than *M*. Therefore, the output of *M* with such an input must be less than *W*(*M*). If we think in terms of runtime as output, then we can conclude that any instruction less complex than *M* will cause *M* to do one of two things: a) run for some time shorter than *W*(*M*), or b) run forever.

So now, dropping all the terminology, we can look at a very curious fact. Assuming there is some upper limit to the complexity of the machines we can build, then there is some number *W* such that any machine which we set into operation must either a) turn itself off in less than *W* seconds, or b) run forever. It is as if there were certain regions of the future that are entirely inaccessible to us.

To make this problem more colorful, imagine that time will continue indefinitely and that you are going to build a time machine which you can set to take you *D* days into the future for any particular *D* that you program into the machine. There would be some future day *D _{w}* such that any attempt to get the machine to take you out past

*D*would end with the machine running forever and taking you infinitely far into the future. The problem would be that you could in no way even

_{w}*conceive*of any finite number of days large enough to reach past

*D*

_{w}.Or, if you don’t like time machines, think of this in terms of time bombs. There is a certain future date after which it can be said with assurance that no time bomb set up by us before the year 2000 can still go off (assuming that the timer on the bomb is some sort of digital device that does not malfunction).10

Let’s go back to the Berry paradox for human beings, instead of for digital computers. One apparent difference between the two is that humans have a certain sort of self-knowledge, which is perhaps not enjoyed by machines. But are we really any better off with out so-called self-knowledge? Don’t we run into precisely the same regress as *M* did when we try to understand what is meant by *u*_{0}, the first number that we can’t name? For to find the first number we can’t name, we have to find out what each name applies to, and take the first number that is left out—but to figure out what each name applies to, we have to figure out what the name *u*_{0} applies to, don’t we?

One way out of this regress is to regard a name like *u*_{0} as being a *second-order name*, so that there will be all the numbers nameable (in under a million words, say) without using the second-order concepts such as “nameable”; then there will be the second-order nameable numbers, such as *u*_{0} or *G(u*_{0}, *u*_{0}, *u*_{0}); beyond all these would be *u*_{1}, the first number not nameable using second-order concepts, etc. But this type-theoretic way out of the Berry paradox is not really satisfying, for one wants to say, “look, by ‘*u*_{0}’ I mean the first number that can’t be named in less than a million words by any means whatsoever, and that includes every order of language that you can invent in under one million words.”

This absolutist view can be visualized as in Figure 51. All the circled numbers are nameable, and *u*_{0} is the first unnameable number. There will probably be runs of nameable numbers further up the line. For instance, “the largest perfect number” might name some extremely large number *P > u*_{0}, and then we would have *P + n* and *P* − *n* nameable as well, for numbers *n* with names not too close to the maximum word limit. Eventually, all the names will peter out and one gets a number *W* that is greater than *every* number nameable in less than the prescribed number of words.

To avoid paradox, one has to accept the fact that the names *u*_{0} and *W* are really *not* names. So the concept of “nameability” is itself really unnameable. The symbols n-a-m-e-a-b-l-e *point* to the concept, but they do not really reach it. In somewhat the same sense, the symbol Ω, which is used to stand for the Absolute Infinite, *points* towards the Absolute Infinite, but cannot really denote it. Just as the Absolute lies beyond any possible description, the notion of *nameable in a lifetime* lies beyond any rational human description.

Is this a final solution to the Berry paradox? Not really. For we are still left with the basic problem of how a sentence like “nameability is an unnameable concept” is meaningful, even though the subject of the sentence is a word that cannot denote any single graspable concept. It is curious how interesting it can be to talk about things that we supposedly can’t talk about!

One conclusion we might draw is that there are two distinct modes of consciousness: the finite and the infinite. As long as I identify with my body and my rational mind, I cannot conceive of my *u*_{0}; but it is not hard to envision my *u*_{0} if I identify with the Absolute. This does not lead to the usual type-theoretic regress, because someone who is merged with the Absolute is in a position to “name” each and every natural number at once.

### RANDOM REALS

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In principle, every real number *r* codes up a countably infinite amount of information: the infinite sequence of digits in *r*’s decimal expansion. In practice, all of the real numbers we ordinarily deal with are actually specified by a finite amount of information. Names such as *π*2, cos 3, log_{10}387 are, in fact, compact and stylized sets of instructions for generating the endless decimal expansions in question.

We will say that a real number is *random* if it has an irreducibly infinite amount of information. That is, a sequence of digits is random if there is no finite way of describing it—no finitely given procedure that can be used to generate the sequence digit by digit. Actually, the word “random” is usually applied only to real numbers that obey certain further conditions having to do with the notion that each nameable subsequence of a random sequence should be random as well.11 But for our purposes, it will be sufficient to equate randomness with unnameability.

The subsection, “Constructing Reals,” contains a historical development of the various sorts of names for real numbers that are used, with particular emphasis on the Greek methods of constructing reals. The subsection, “The Library of Babel,” presents and analyzes the idea of a Total Library containing all possible books. In the somewhat technical subsection, “Richard’s Paradox,” a construction discovered by Jules Richard is examined. This construction seems to produce a random real by diagonalizing over the set of reals having names in the Total Library; it raises some of the same issues as did the Berry paradox. In the final subsection, “Coding the World,” the question of the physical existence of random reals is investigated.

#### CONSTRUCTING REALS

In this section it will be useful to think of real numbers as points on an idealized continuous line. The points called 0 and 1 must be marked arbitrarily, but from then on the correlation between real numbers and points on the marked line is more or less automatic.

The ordinary representation of a real number as an endless decimal expansion can be viewed as a description of an *infinite* procedure for locating a particular point (or infinitesimal neighborhood) on the marked line. If we allow ourselves the use of various standard curves and surfaces, then there are many real numbers for which there is a *finite* procedure for locating the corresponding point on the marked line.

Assuming that we have a compass, the point called 2 can be found in a finite amount of time by drawing the circle with its center at 1 and radius equal to the interval between 1 and 0. It should, perhaps, be pointed out that, in practice, the perfect drawing of this circle would actually be an infinite process. For even if our lines had no thickness and our compass’s drawer and stabber were points, there would still be the difficulty of managing to put the stabber exactly through the point 1 and contriving to set the drawer down exactly on the point 0. In practice, matching up two points is an infinite process of repeatedly reducing one’s error—a feedback loop. To avoid this objection to the finiteness of the construction of 2, one may as well assume that perfect circles with specified centers and radii will spring into existence on command, as will perfectly straight lines through specified pairs of points. This, after all, is the content of Euclid’s third and first postulates, which do not actually mention “rulers and compasses.”

The upshot is that given 0 and 1 we can, without idealizing too much, find the point corresponding to the real number 2 with infinite precision in a finite amount of time. The same is true of any real number whose decimal expansion terminates or repeats, for such a real number is rational, and every rational point on the marked line can be found with ruler-compass constructions.

As an example, I show how one might find the point corresponding to the real number 2.40000…. This is the same as so the core of the construction is to find a line segment of length This is done in Figure 53 by means of a trick that uses similar triangles to divide a given interval (in this case a copy of the interval between 0 and 1) into five equal pieces.

Using ruler and compass we can not only construct any rational number, but also construct any number obtained from the rationals by combining the operations of addition, subtraction, multiplication, division, and the taking of square roots. (Recall that in the section “From Pythagoreanism to Cantorism” I showed how to take square roots with ruler and compass.) So a point on the marked line corresponding to, say, or can be found with infinite precision in a finite number of time if we are allowed to call circles and lines into existence as desired.

The three famous problems of antiquity have to do with the question of which ruler-compass constructions are possible. I refer here to the problems of Duplicating the Cube, Trisecting the Angle, and Squaring the Circle. The first and third of these problems are of special interest here.

In the Duplicating the Cube problem, one wants to construct a cube that has twice the volume of a given cube. It is not hard to see that this is tantamount to finding a finite method for locating on the marked line the point corresponding to the real number

In the Squaring the Circle problem, one wants to construct a square that has the same area as a given circle. This amounts to being able to find a finite method for locating on the marked line the point corresponding to the real number or, what is equivalent since we can square and take square roots with ruler and compass, the real number *π*.

Someone brought up on the modern theory of real numbers is prone to feel that simply writing down and understanding the symbols solves the problem of Duplicating the Cube. For, after all, once you understand that means “the real number whose cube is 2,” you can set out to determine digit after digit of the decimal expansion of

If you like, you can locate the points corresponding to each of the initial pieces of the endless expansion of and after an infinite amount of time you will reach the limit point that represents

But this is nothing the Greeks did not know about. They were fully aware that one can find points that better and better approximate the location of on the marked line, or that given a marble cube one can find larger marble cubes that come closer and closer to weighing twice as much as the original cube.

By the process of trial and error we can always come closer and closer to finding a continuous magnitude with some specified property. The *axiom of continuity* (first explicitly introduced by Dedekind) asserts that there is a single right magnitude that exists as the limit of any such process. In modern times we have found it convenient actually to identify continuous magnitudes with such processes—a particular continuous magnitude is represented by the specific process of trial and error coded up in the decimal expansion of the corresponding real number.

But when the Greeks posed the problem of duplicating the cube, they wanted a *finite* method of constructing a line segment with length *exactly* equal to They were suspicious of infinite processes, holding that no such process could legitimately be regarded as completed, and in the absence of a finite construction of the location they might even have questioned whether any such perfectly right location really exists at all.

The first person to come up with a finite method for constructing was Archytas, a member of the Pythagorean sect. Archytas was a friend of Plato, and is said to have once prevented Dionysius from killing Plato (Dionysius being Plato’s school master). Archytas is also credited by Aristotle with having invented a model bird that flew and a special sort of baby rattle—all of this in the fourth century B.C.

Archytas’s method for finding the cube root of two involves considering the intersection point of a torus, a cylinder, and a cone. One starts with the circle with diameter *OC* equal to 2 (Figure 57A). On this circle, find a point *B* such that the chord *OB* has length 1.

We generate the torus by rotating a copy of this circle about a line through 0 and perpendicular to *OC.* We generate the cylinder by moving the circle perpendicular to itself. We generate the cone by extending the chord *OB* until it intersects a line passing through *C* perpendicular to *OC*, and by then rotating the triangle formed about this line.

If these three surfaces are placed so that the original circles match, then there will be a point *P* where all three surfaces intersect. Directly beneath *P* will be a point *X* on the original circle, and the distance *OX* equals In order to vizualize where *P* is, I have drawn on the cylinder the curves where the cylinder would intersect the cone and torus. *P* lies at the intersection of these two curves.

The reason that *OX* is has to do with the fact that this particular construction leads to a right triangle of the kind drawn in Figure 57E. Because all of the triangles in this figure are similar, we get the continued proportion that leads to the solution of the Duplicating the Cube problem.

The Duplicating the Cube problem arose when the oracle at Delos told the Delians that a plague they were currently suffering would be lifted only if they would double the size of the temple altar. The Delians were clever enough to realize that the volume of *any* three-dimensional object is doubled if each of its dimensions is increased by a factor of and thus the problem was born.

The Greeks seem to have realized quite soon that no ruler-compass construction would suffice to construct cube roots. There are a number of recorded solutions to the Duplicating the Cube problem, all of them involving some sort of higher-order curve or surface. A thorough catalogue of these solutions appears in T. L. Heath’s classic work, *A History of Greek Mathematics*, written during the First World War.

In his preface, Heath quotes a remark of Plato’s on the Duplicating the Cube Problem that must have seemed particularly apt during those war years: “It must be supposed, not that the god specially wished this problem solved, but that he would have the Greeks desist from war and wickedness and cultivate the Muses, so that, their passions being assuaged by philosophy and mathematics, they might live in innocent and mutually helpful intercourse with one another.”12

It was not conclusively proved that cannot be constructed using only lines and circles until early in the 1800s, but, as I have mentioned, the Greeks suspected this from the very beginning, and took the Duplicating the Cube problem as an impetus to move on to more complicated, *but still finite*, methods of construction.

It should be noted that Archytas’s method is a fairly natural extension of ruler-compass methods. Ruler-compass methods allow one to: i) form the trail of a point moved in any fixed direction; ii) form the trail of a point rotated about any other point. To get the cylinder, torus, and cone of Archytas we need only be allowed to: i) form the trail of any plane curve moved in any fixed direction; and ii) form the trail of any curve (ie., plane figure) rotated about any line.13

As we know since Ferdinand Lindemann’s proof of 1882, the number pi is transcendental, which means that it can never be constructed by taking intersections of simple (algebraic) curves and surfaces a finite number of times. One might be inclined to say that pi can be constructed simply by rolling a circle of diameter one through one revolution, but this seems to involve the notions of motion and of time, which are perhaps extra-geometric. Actually the notion of rolling the circle can be statically represented by the helix in Figure 58. This is a helix that moves up just as fast as it moves around the cylinder, making a 45 degree angle with the cylinder’s generators at each point. Evidently, the vertical change produced during one complete revolution will be equal to the circumference of the circle.

A somewhat more constructive process for getting pi in a finite amount of time is due to Archimedes, and is illustrated in Figure 59. The idea is that one takes an Archimedean spiral, obtained by rotating a line at a constant rate while a point moves out along this line at some other (not necessarily related) constant rate. If the spiral starts at *O* and completes its first turn at *P*, then one can draw the line *PT* tangent to the spiral at *P* and the line *OT* perpendicular to *OP* at *O*, and the point *T* where these two lines cross will determine a distance *OT* equal to pi times *OP*.

The Greeks also used a much more contemporary method for finding approximations of pi: the method of exhaustion. Originated by Antiphon and perfected by Archimedes, this method consists simply of inscribing and circumscribing polygons with more and more sides upon a given circle. One can, in general, compute the perimeter of any polygon, so that by finding the perimeter of, for instance, a 96-sided regular polygon that just fits inside a circle with diameter one, it is possible to get a good approximation of pi. By precisely this method Archimedes showed that and there is, in principle, no limit to the accuracy obtainable by this method (although, of course, infinite precision will never be obtained in a finite amount of time in this manner).

I have illustrated Archimedes’s method of exhaustion in Figure 60. The idea is that one starts with a regular inscribed hexagon at stage 1, and that by repeatedly bisecting the subtended arcs of the circle one gets to a 3 · 2*n*-gon at stage *n*. The length *S _{n}* of the side of a 3 · 2

*n*-gon inscribed in a circle of radius is given by the recursive formula displayed in Figure 60. To see how the formula is obtained one can think of the figure as illustrating the general transition from

*S*to

_{n}*S*, and by using the Pythagorean theorem twice the formula can be obtained. Given the side

_{n+1}*S*of the regular 3 · 2

_{n}*n*-gon inscribed in a circle of radius we can approximate

*π*by the perimeter

*P*= 3 · 2

_{n}*n*·

*S*of this polygon.

_{n}In a sense, the three formulas in Figure 60 give a finite description of pi, and in a sense they do not. On the one hand, the three formulas can be used to compute *P _{n}* for arbitrarily large values of

*n*, and for large enough

*n, P*will be arbitrarily close to pi. On the other hand, these three formulas will never give the exact point pi on the number line after any finite length of time.

_{n}Zeno’s bisection paradox brings this distinction into sharp relief. If you are at 1 on the number line, there are two ways to get to 2. You can use the finite procedure of moving one unit all at once; or you can use the infinite procedure of moving unit, then then etc. If we ignore the possibility of there being distinct points an infinitesimal distance apart from each other, then it is evident that these two procedures lead to the same point. This fact is usually represented by the equation Zeno viewed this as paradoxical. He assumed *a priori* that no actual infinity could exist, so that no infinite process could be regarded as completed; therefore the equivalence between a finite and an infinite process seemed impossible.

As I have mentioned before, the ordinary decimal expansion of a real number actually represents an infinite process for finding a point corresponding to the number in question. Although infinite, this process is quite definite, and we feel comfortable with it. It should be realized, however, that for many of these real numbers there is no alternate finite process at all for finding the length described.

Take, for instance, the number *L* = 10−1 + 10−2! + 10−3! + 10−4! + 10−5! + . . = .110001000000000000000001000000000000000000

00000000000000000000000000000000000000000000000000

00000000000000000000000000010.… (Recall that *n*! means *n* factorial, the product of the numbers one through *n*). In 1844 Joseph Liouville proved that this number is transcendental, meaning that *L* is not the root of any polynomial with rational coefficients. There are many transcendental numbers, but *L* was the first number for which anyone was able to *prove* that it is transcendental. In fact, *L* is artificially tailor-made to make just such a proof possible.14

Since *L* is transcendental, it can never be found by a method such as Archytas’s technique of intersecting algebraic curves and surfaces. Since *L* is highly artificial, it is most unlikely that a segment of length *L* can ever be found by any other finite method … such as the spiral and helix constructions of *π*. So *L* is a number for which the corresponding length can only be found by the sort of infinite process questioned by Zeno.

One could, of course, idealize a bit more than previously and imagine a device–call it a Summer–that will construct lengths corresponding to numbers such as *L*. The Summer geometrically sums up any infinite series in one second. Thus, given a decmal expansion .*r*_{1}*r*_{1}*r*_{3}…, the Summer moves a pointer to .*r*_{1} in the first second, finds .*r*_{1}*r*_{2} in the next second, fnds .*r*_{1}*r*_{2}*r*_{3} in the next second, and at the end of one second, the Summers pointer indicates the point corresponding to the decimal series .*r*_{1}*r*_{2}*r*_{3}….

This is really not *so* great an idealization–recall that we already had to idealize in order to talk about ruler-compass constructions. In a sense, when you move a pointer from 1 to 2 on the number line in one second, you are acting as a Summer on the series This, incidentally, was part of Aristotle’s answer to Zeno’s bisection paradox: Since we do indeed act as Summers when we move in time, there is nothing fundamentally contradictory about them.

One must be careful, however, not to feed a divergent series into the Summer, or it will break. If, for instance one gives the Summer the Grandi series to work on, then it will shake itself to pieces. The Grandi series is the series 1 − 1 + 1 − 1 + 1 − 1+… Looked at one way this is (1 − 1) + (1 − 1) + … = 0; but looked at another way, we have 1 + (−1 + 1) + (−1 + 1) + … = 1!

Grandi discovered this series in 1703; he claimed that God must have used a technique based on this series in order to create Something from Nothing, and thus get the cosmos going.15 This is not really as insane as it sounds—a more sophisticated way of saying something similar would be to claim that the cosmos is a moiré, an interference pattern produced by a wave function out of phase with itself.

But in any case, if one feeds the Grandi series into the Summer, the pointer is going to twitch back and forth between 0 and 1 infinitely many times at a speed that reaches infinity as the second runs out. Unless some special provision (e.g., “point to 1 when in doubt”) has been built into the Summer, there is simply no reason why it should point to any one point instead of any other point at the end of a second … so I would prefer to say that it breaks.

The question of what a Summer does with divergent infinite series has been thoroughly discussed by philosophers of science as the problem of the Thompson Lamp.16 There is no problem with the action of a Summer on a convergent series such as the decimal expansion of a real. The increments of motion become smaller and smaller, and at the end of the second the pointer has smoothly come to rest at a particular point. When you hold a hard pencil between your two fingers and let the tip bounce on a table, you see a motion something like that of the pointer of a Summer acting on a convergent series. In principle, the pencil point bounces an infinite number of times, but the total distance travelled can be computed to be finite, and the total time elapsed is finite as well. In particular, if the pencil point starts out at a height of one inch above the table, and each time it bounces back to the height from which it has dropped, then it will fall a total distance of and the time elapsed can similarly be seen to be finite.

From now on, we will say that a real number *r* has been finitely described if there is some finite description of how to generate the decimal series expansion *R.r*_{1}*r*_{2}*r*_{3} … of *r*. In general, such a finite description will consist of some general set of instructions that, when applied to any natural number *n*, will institute a process terminating with the evaluation of the digit *r _{n}*. Any such description of a general function giving an

*R*and an

*r*for each

_{n}*n*can be fed into the Summer to locate a particular point on the real line.

There is, of course, a little vagueness in the words, “finite description” and, as we will see in the “Richard’s Paradox” subsection, this vagueness leads to complications like those that came out of the Berry paradox concerning the words “describable in less than a billion words.”

#### THE LIBRARY OF BABEL

Eventually we want to try to find a random real number—that is, a real number that has no finite description whatsoever. But before going on with this project, we need to get a clearer notion of the set of all possible finite descriptions.

It is initially easier not to be concerned with questions of meaningfulness, and to view *any* string of symbols whatsoever as a “possible description.” The set of all finitely long strings of typographic symbols is, in any case, an intrinsically interesting set.

Related sets have been discussed in “The Universal Library” by Kurd Lasswitz17 and in “The Library of Babel” by Jorge Luis Borges.18 The Borges story is a first person account narrated by an inhabitant of a fantastic library consisting of an apparently endless number of hexagonal rooms lined with identically bound volumes. Each volume has 410 pages consisting of forty lines of eighty symbols each. Most of the books seem to be meaningless jumbles of letters.

The narrator and his fellows spend their whole lives wandering in this library, ceaselessly speculating on what it all means. Some believe that every single book has meaning—if not in Spanish, then in English or Hungarian; if not in any known language, then in a language of the future or in code. But a book consisting, say, of 410 pages of b’s seems to be meaningless under *any* interpretation; and the narrator draws the conclusion that the library actually includes every possible book-length string of symbols.

He feels that the library includes everything, and one of those marvellous Borgesian lists ensues:

Everything: the minutely detailed history of the future, the archangels’ autobiographies, the faithful catalogue of the Library, thousands and thousands of false catalogues, the demonstration of the fallacy of those catalogues, the demonstration of the fallacy of the true catalogue, the Gnostic gospel of Basilides, the commentary on that gospel, the commentary on the commentary on that gospel, the true story of your death, the translation of every book in all languages, the interpolations of every book in all books.19

In a sense a library like this is useless. Randomly selecting a book from the Library of Babel is equivalent to sitting down and randomly typing 410 pages. Even if you were, by some miracle, to find a book in the Library that seemed to provide a solution to Cantors Continuum Problem, you would have to check very carefully to make sure that you had not obtained one of the thousands of false versions of this book … and even if your book seemed to be without error, it might be possible to find another error-free book that provided a very different solution to Cantor’s Continuum Problem. Looking at the titles of the books would be of no help, for a book called *The Continuum Problem* might turn out to be about, say, astral travel.

The narrator of “The Library of Babel” actually claims too much for his library in the quote above. The detailed history of the future would (unless the world is to end quite soon) probably not fit into any 410-page volume; and the catalogue of the immense Library of Babel would certainly not fit into any 410 pages. To be really certain of having everything in one’s library, one should allow books to have any finite length, no matter how long. But then one’s Library becomes infinite.

In a belated urge to economize, Borges limits the symbols used in the books in his Library to twenty-five: the lower-case letters of the alphabet (excluding h, k, w, and x), the period, the comma, and the space. But it is too late for economy. Each book in his Library has 1,312,000 slots (= 410 × 40 × 80), so there are 251,312,000 *≈* 102,000,000 books in the Library of Babel … barring repetitions.

Kurd Lasswitz arrives at the same figure for the number of volumes in his Universal Library, and to point out the largeness of this number he remarks that if his books were placed side by side it would make a shelf about 101,999,982 light-years long. The fact is that there are so many books in this Library that the number of light-years of books is not substantially smaller than the number of books.

As I mentioned above, if we allow our books to be arbitrarily long, then there are infinitely many books in what could be called the Total Library. We can see that the Total Library is infinite, since for each natural number *n*, there must be a book consisting of *n* repetitions of the word “yam.” Thus all of these are books in the Total Library: yam, yamyam, yamyamyam, ad inf.

Given that the Total Library is already going to be infinite, there is not much point in trying to cut corners on the number of symbols used in the books. A book such as *Infinity and the Mind* uses something like three hundred typographical symbols, but to keep the rest of the discussion manageable, we’ll limit ourselves to the seventy-five most basic symbols: the space, the lower and upper case Roman alphabets, the digits 0 through 9, the apostrophe, comma, dash, semicolon, colon, period, exclamation point and question mark, the left and right quotation marks, and the left and right parentheses.

Is there any danger that the Total Library is *uncountably* infinite? If you are careless, you might think that it is, reasoning as follows: I can create an arbitrarily long finite book using the seventy-five basic symbols by choosing one of these symbols ω times. This can be done in 75 × 75 × 75 ×… = 75ω = *c* ways; therefore, the Total Library has the uncountable cardinality of the continuum.20

The flaw in this argument is that what has really been calculated is the number of books of length ω, rather than the number of books having length less than ω. The Total Library has cardinality ℵ_{0}, and we prove this by constructing a one-to-one map, called CODE, from the set of all finite books into the set of natural numbers. (This proves that the cardinality of the Total Library is ≤ ℵ_{0} and since the “yam” argument shows that this cardinality is ≥ ℵ_{0}, the rules of transfinite arithmetic imply that the cardinality in question is *equal* to ℵ_{0}.)

To set up the CODE map we start by assigning a ditigal code to each of the seventy-five basic symbols.

-1 | y -28 | X -56 |

a -2 | z -29 | Y -57 |

b -3 | A -31 | Z -58 |

c -4 | B -32 | 0 -59 |

d -5 | C -33 | 1 -61 |

e -6 | D -34 | 2 -62 |

f -7 | E -35 | 3 -63 |

g -8 | F -36 | 4 -64 |

h -9 | G -37 | 5 -65 |

i -11 | H -38 | 6 -66 |

j -12 | I -39 | 7 -67 |

k -13 | J -41 | 8 -68 |

1 -14 | K -42 | 9 -69 |

m -15 | L -43 | ‘ -71 |

n -16 | M -44 | , -72 |

o -17 | N -45 | - -73 |

p -18 | O -46 | ; -74 |

q -19 | P -47 | : -75 |

r -21 | Q -48 | . -76 |

s -22 | R -49 | ! -77 |

t -23 | S -51 | ? -78 |

u -24 | T -52 | “ -79 |

V -25 | U -53 | “ -81 |

w -26 | V -54 | ( -82 |

x -27 | W -55 | ) -83 |

We are careful to use only code numbers with no zeros in them, reserving zero for a different purpose. To code up a given string of symbols, we replace each symbol by its code number, put zeros in between the code numbers so we can tell them apart, and then stick everything together to get a big natural number.

Using the zeros as spacers makes it possible to decode any code number.

Of course, most numbers don’t code up anything at all under this system. Thus, 235,794 simply doesn’t code up anything. But the point is that every book in the Total Library *is* coded up by some finite natural number.

The reader may enjoy checking that the number CODE (*Moby Dick*), which codes up the text of *Moby Dick*, starts out 33020140140101506010390220901502060140760 …

It will be convenient later on to think of the ℵ_{0} books in the Total Library as being listed *B*_{0}, *B*_{1}, *B*_{2}, … in order of the size of their code numbers. In effect, this arrangement lists all the books one symbol long, then all the books that are two symbols long, and then the three-symbol-long books, and so on.

Coming back to the coding process for a minute, notice that there is, in principle, no reason why a child could not be taught from the beginning to read book codes instead of books. You’d teach him that 2, 3, 4, … is the alphabet (pronounced hay, bee, sea, …), teach him to use 76 at the end of a sentence, teach him always to separate his symbols by zeros, and so on.

For a person taught to read in this manner, the Total Library would simply be an infinite subset of the set of natural numbers. “Have you read 3, 702, 102, 025, 011, 023, 028, 071, 022, 010, 490, 201, 101, 603, 017, 026?” “Yes, I liked it even more than his 54.” The point is that there is nothing sacred about our particular letter symbols. What is essential in a book is the overall pattern these symbols form.

A curious thought arises here. Suppose that we gave the code numbers for all the books in the Library of Congress to some blob from outer space. To the extent that the meaning of any word can be explained, the explanation of all the recurring symbol patterns will be provided by other symbol patterns in this welter of data. Would the alien ever be able to figure out what the books were about? Even if he couldn’t understand them in the usual sense, would he be able to appreciate the abstract patterns of symbols formed by the classics?

Note that these questions would equally well arise if the alien were given the books in English. The point of considering the *codes* of the books in this example is just to bring out the fact that given a text of ours, an alien would have only the text’s abstract structure to work with.

With this in mind, the scientists interested in extraterrestrial communication have devised certain very simply patterned messages to be beamed into outer space. A recent spacecraft also carried some more complex information patterns for the aliens, notably a recording of Chuck Berry’s “Johnny B. Goode.” Reading off the digital coding of this song, the aliens may notice certain regularities of pattern, certain mathematical progressions.

A song is a curious sort of information pattern in that it has no real content—it is appreciated simply for its form. One might be tempted to think that we can somehow teach aliens what our words mean, perhaps by means of chemical formulae, etc. Still, there seems to be a large portion of our language experience that can be taught only by direct demonstration. “There, feel that, drink … that’s *water*, Helen, *water*.”

It may be that the aliens will enjoy our messages only sensuously, in the way we enjoy music and abstract art. Then again, they may read meanings peculiar to their own world-views into our messages.

The discussion of aliens looking at our books is quite relevant to the question of when a string of symbols *names* a real number. For some, the two symbol string “pi” names a definite real number. Someone else might prefer the longer name, “pi is the ratio of the circumference of a Euclidean circle to its diameter.” An individual unfamiliar with mathematics might require this last definition to be amplified to a complete treatise on plane geometry.

Presumably, a complete enough treatise would enable any type of thinking creature to derive and use the formulae for pi set out in Figure 60, even if this creature had no idea of the kind of visual and tactile experiences that humans associate with “circles.”

A different approach to naming pi would have been simply to start with these formulae and some sort of an explanation of how to use them. Alternatively, one could use the description of pi that is usually proved towards the end of a second-year calculus course: “Pi is the limit of the infinite series

In a sense, digital computers are the only aliens that we are able to talk to at all, so they are often taken as a standard of nameability. Every large general-purpose computer is fundamentally the same (they are all “universal Turning machines”), so we can talk about a general computer *C* without having to be too specific. One can formally say that the book *B _{i} names* the real number

*r*=

*K.e*

_{1}

*e*

_{2}

*e*

_{3}

*e*

_{4}… provided that feeding

*B*into

_{i}*C*throws

*C*into such a state that if zero is fed into

*C, C*prints out

*K*, and if any

*n*greater than zero is fed into

*C*, then

*C*prints out the

*n*th. digit

*e*of the decimal expansion of

_{n}*r.*

In the next subsection we will use the phrase “*B* names the real number *r*” in several ways. If the specific way just given is meant, then we will sometmes say “*B* is a *C*-name for the number *r*” to emphasize this.

#### RICHARD’S PARADOX21

Let *M* be some type of being: a computer, a human, the human race as a whole, a thinking galaxy, or God Himself. We say that the finite string of symbols *B* is an *M*-name for the real number *s* exactly when *M* is able to give the number *s* on the basis of the information in *B.* As was discussed in “Constructing Reals,” we may know the value of a real number, even though we cannot give the full infinite decimal expansion all at once. We will say that *M* is able to give us the real number *s*, provided that *M* is in a position to give the *n*th digit of the decimal expansion of *s* for any desired *n.* Thus, I say that I am able to give the real number pi, not because I have the entire decimal expansion in my mind at once, but, rather, because for whatever *n* you mention, I can, in time, respond with the *n*th digit of the decimal expansion of pi. I can do this because I have a certain technique for computing more and more digits of the expansion of pi. For me, a string of symbols describing such a computation technique serves as a name for the real number pi.

Now, let us fix our attention on some one particular *M* and consider the set *E _{M}* of all the real numbers that have a finite

*M*-name somewhere in the Total Library. It seems that since we can find a translation function Trans

*that maps the countable Total Library onto the set*

_{M}*E*must be countable. Now, the diagonalizing technique studied in Excursion I shows how to find a real number different from every member of any given countable set of reals, so it seems that for any

_{M}, E_{M}*M*, there will be real numbers that do not have any finitely long

*M*-name; these reals might be called

*M-random*reals.

The question of whether or not there is a best *M* comes up here. That is, is there a sort of ultimate *M* such that if a real number *s* has any finite name at all, then it has an *M*-name? If this is indeed the case, then we can think of the *M*-random real numbers as being random in the absolute sense of having no finite description at all.

Since it is the nature of infinite sequences of digits that really concerns us here, it will be no great loss if we restrict our attention to the real numbers between zero and one—that is, the real numbers with nothing to the left of the decimal point.

Returning to the *M*’s, it is simpler to think of them behavioristically. An *M* is something that turns natural numbers into real numbers, and if two *M*’s behave the same we regard them as identical. So we can identify an *M* with a certain list of real numbers, which can in turn be coded up by a single real number.

In general, for a given *M* and a given natural number *n*, we can define Trans_{M}(*n*) to be the real number .*e*_{n1}*e*_{n2}*e*_{n3}*e*_{n4} … given by the definition.

In terms of behavior, *M* is given by the doubly infinite square array of all the *e _{nk}.* By means of a certain sort of shuffling, we can fit all of these

*e*into a single

_{nk}*ω*-sequence, which can be regarded as a real number that might as well be called

*T*or .

_{M}*m*

_{1}

*m*

_{2}

*m*

_{3}.…

There is another interesting number obtainable from the square array given. This is the diagonal number *d*_{M} = .*d*_{1}*d*_{2}*d*_{3} …, defined so that

Now, *d _{M}* is different from every one of the Trans

_{M}(

*n*), which is to say that

*d*

_{M}is

*M*-random and has no

*M*-name.

A little thought shows that *d _{M}* can be defined directly from

*T*since

_{M}*e*is always in the 2

_{nn}*n2*− 2

*n*+ 1 place of

*T*. So we could also use the right-hand definition of

_{M}*d*The dependence of

_{M}.*d*on

_{M}*T*can be expressed by saying

_{M}*d*=

_{M}*f*(

*T*

_{M}).

We will say that a naming system *M* is *closed* if whenever *M* has a name for some real number *s*, then *M* also has a name for real numbers that have simple definitions in terms of *s.* In particular, we say that *M* is closed only if whenever *M* names a real number *s*, then *M* also names the real number *f*(*s*), meaning that if *M* names the code *T _{N}* of some naming system, then

*M*will also name the diagonalization

*d*of this naming system. Given the definition above of

_{N}*d*in terms of

_{M}*T*, we can see that any naming system that would naturally be adopted by a rational being would be closed in this sense.

_{M}Now, consider a closed naming system *M* · *M* cannot name the diagonal number *d _{M}*, since this number is constructed to differ from every real number with an

*M*-name. If

*M*has a name for

*T*

_{M}, then since

*M*is closed,

*M*will also name

*d*

_{M}—but this is impossible. Therefore,

*M*does not have a name for

*T*. In general, no closed naming system can name the real number

_{M}*T*that codes up this system.

_{M}This fact was first discovered by Jules Richard in 1905. Richard was a French high school teacher at the time. He formulated the fact that no closed *M* names *T _{M}* as a paradox by taking his naming system to be a given and evident universal relation. That is, he assumed that “

*B*is a name for

*s*” is a relation that is already perfectly clear. But if this relation is already perfectly clear, then the real number

*T*coding up all the nameable reals is clearly defined, and then

*d*(the diagonalization) is clearly defined. But

*d*cannot be nameable since it differs by construction from every nameable real. So, if we assume that the relation of naming is clearly named by the word “name,” then we can name a number

*d*that differs from every number that we can name. Such is Richard’s paradox.

Richard himself was able to see that the way out of the paradox is to deny that the *M* one has in mind can name *T _{M}.* As he puts it, the diagonal number is really only named by

*M*if the translation code

*T*is totally defined, “and this is not done except by infinitely many words.”22

_{M}Is there any deeper significance to the fact that no closed naming system *M* names its translation code *T _{M}*? Looked at quite formally, we are simply saying

that when certain real numbers *T*_{M} are decomposed into countably many real numbers in the way indicated above, then the original number does not appear among the countably many real numbers obtained. This, of course, applies only to numbers *T _{M}* that code up a closed system of naming. It is worth noting, incidentally, that if

*T*does code up a closed system, then breaking down one of the new numbers like Trans

_{M}_{M}(3) should not lead to any numbers not already present after the first breakdown.

If we think of *M* as a human being, then we can imagine the *n*’s as being books made of words, and think of Trans_{M} as being the process of translating each book into a real number. There can be no book that gets translated into the real number *T _{M}* coding up all of

*M*’s activity.

We can make this more colorful. The code numbers can be thought of as books made of *words*, and we can perhaps think of the real numbers named by these words as being *ideas.* The idea of pi embodies the whole infinite decimal expansion in a simple pattern. In general, a real number is something like an idea, because it has a definite existence as a mental (as opposed to physical) object, yet nevertheless provides a normative standard for concrete approximations (namely, the initial segments of the infinite decimal expansion). So, we might say that Richard’s argument demonstrates that no human being can give a finite description of how he turns words into ideas.

This is quite similar to the moral already drawn from the Berry paradox—that no finite scheme can capture the essence of how one connects the real and the ideal, the physical and the mental, language and thought.

But wait. Even though *M* has no finite name for *T _{M}*, couldn’t there be a better naming system

*M** that

*does*have a name for

*T*? There seem to be two alternatives. On the one hand, it may be that

_{M}*T*is absolutely unnameable, in the sense that no finitely given system

_{M}*M** can name

*T*In this case,

_{M}.*M*must be infinitely complicated. On the other hand, it may be that there is some finite way of giving

*M*and

*T*within a higher system

_{M}*M**. In this case,

*M*is incomplete. So we can conclude that

*any system for naming real numbers must either be infinitely complicated*(so that

*T*is itself random in the absolute sense)

_{m}*or incomplete*(so that

*T*does eventually admit of a finite description).

_{M}The real motive in taking up Richard’s paradox was to try to find a real number that is absolutely random and answers to no finite description. If we believe that there is such a thing as a maximal naming system *U*, then it cannot be that *U* can be improved to a naming system *U** that names everything *U* names, and names *T _{U}* as well. So if we believe that there is a maximal naming system

*U*, then we know that there is a random real number—the translation code

*T*If we accept the relation “

_{U}.*B*names

*s*” as being meaningful without further specification, then we are really thinking in terms of some maximal system

*U*. But there is some question as to whether this is a legitimate way to think.

One could be rather hard-nosed about it and deny that a naming system *M* exists unless it has been exhaustively specified by various rules and schemata. Such an *M* is basically a finite thing, and can always be improved upon to get a better *M** that also describes *T _{M}.* Now, this kind of process can be continued indefinitely without ever reaching a stopping point. It is a bit like the way in which we can always find a greater natural number without ever getting to an actually infinite number.

So a Richard-style argument will give us an irreducibly infinitely complex real number *T _{U}* only if we are already willing to accept the existence of a somewhat transcendental relation of “naming.” Such a

*U*would be embodied in, let us say, a God who was an English-speaking mathematician. But all we are doing then is assuming one infinity to get another one.

#### CODING THE WORLD

One can conceive, ideally, of a set of facts that would enable one to answer every possible question about our universe. The question that concerns us here is whether or not the smallest, most efficient such complete description of the world is finite or infinite.

Whether or not the world itself is infinite to begin with makes a difference, of course. But this was already discussed in the “Physical Infinities” and “Higher Physical Infinities” sections. Here we are concerned with distinguishing between the following three cases: 1) The universe is totally finite, and thus admits of a finite complete description; 2) The universe is in some respect infinite, but nevertheless is completely specified by some finite set of facts; 3) The universe is infinite and cannot be completely described by any finite collection of sentences.

Case 1 is the situation where space and time are finite and quantized. Here the universe has only a finite extent, and space-time is grainy, so that only a finite number of possible space-time locations exist. A complete description of the universe could be given by specifying what was to be found at each of the finitely many space-time locations.

A universe such as this is something like a picture made of individual light bulbs, such as one sees in displays at Times Square. Or, one might think of the universe as being a large, but finite, four-dimensional Go board with white stones for matter and black stones for antimatter.

I might mention in passing that the German mathematician Eduard Wette believes that the universe is totally finite in the way just described, having well under 101010 space-time locations. He concludes from this that any mathematical talk about numbers greater than 101010 is meaningless, and even inconsistent. He has repeatedly tried to use this insight to fashion a convincing proof that all of traditional mathematics is contradictory.23 Needless to say, Wette’s ideas are unpopular among mathematicians.

Case 2 is the rationalists’ dream. Here we have an infinite universe whose very essence is somehow captured by a finite set of facts and natural laws. Science continually works to approach this situation by finding laws that account for and summarize a wide variety of individual cases. Once we know about the proton-neutron-electron model of the atom, then the table of elements becomes very easy to understand.

An extreme example of this sort of process appears in Eddington’s *Fundamental Theory*.24 Here Eddington tries to derive such physical constants as the mass of the electron and the radius of the universe from certain *a priori* theoretical considerations. His efforts must be judged largely unsuccessful, but the idea of finding a few key facts and laws that account for everything is still an attractive one.

One could, of course, question whether any finite theory could somehow predict the mass of infinitely many stars, or even the arrangement of the blades of grass in a lawn. Actually, there is a certain sense in which no finite theory can exhaustively describe an infinite world. Gödel’s Incompleteness Theorem, which will be studied in the next chapter, states that no finite theory can predict all of the true facts about natural numbers. Now, if the universe is infinite, then it embodies the full set of natural numbers, so Gödel’s Theorem seems to say that for any given finite theory of the universe, there are certain facts having to do with sets of physical objects that cannot be proved by the theory.

But let us set this difficulty aside for now and look at a more concrete way in which a universe might fail to answer to any finite description. We will consider a universe that continues expanding forever after an initial singular state. Our own universe may very well be like this. Given an infinite future, with no future collapses to rub everything out, might look for an irreducible infinity in the form of a random sequence.

Suppose you started flipping a coin and wrote down a 1 for every “heads” and a 0 for every “tails”. If you put a decimal point in front of it all, something is generated that looks like it might be a random real number, say, .0110010100001011…. But there is the problem that you will not be around forever, so you will not be able to produce an infinite sequence of digits.

To avoid this, you might build a coin-flipping machine. To keep the machine running, you supply it with a couple of repair robots, who are also capable of repairing and even building copies of themselves. In order that these three friends don’t run out of energy or raw materials, we put them in a spaceship that zooms around the universe scooping up matter and converting it into energy and the desired elements. If we have an everlasting universe with an infinite amount of matter in it, then there is no theoretical reason why such an immortal coin-flipper could not be set up.

If the coin-flipper is unbiased, it is at least logically possible that nothing but heads will come up from now on. But we would expect it to be more likely to produce a sequence of zeros and ones answering to no finite description. A universe in which at least one such random coin-flipper existed would not have any finite description.

It may very well be that no one will ever bother building such a complicated machine. Isn’t there some simpler sort of physical choice mechanism?

Consider a hydrogen atom, consisting of an electron circling a proton. The atom can exist in various energy states. In general, it moves to higher energy states by absorbing photons, and passes to lower energy states by emitting photons. Suppose that we watch a particular hydrogen atom, and at the end of each second we write down 1 if the atom has emitted a photon during that second, and write down 0 otherwise.

There is no reason why a given hydrogen atom may not survive intact for the rest of time, but there is a problem here in that we will not be around for the rest of time to watch it and mark down zeros and ones.25 At first one is inclined to feel that *whether or not anyone looks* the atom will, during any given second, either emit a photon or fail to do so. So one is tempted to believe that if time goes on forever, then each hydrogen atom in a certain sense already embodies or labels an infinite sequence of zeros and ones.

Randomness is not usually an axiom explicitly assumed in quantum mechanics, but there is a strong feeling that the behavior of a hydrogen atom is, in principle, unpredictable. So we would expect most of the hydrogen atoms to be generating random real numbers. (There is one theorist, Paul Benioff, who has tried extending Quantum Mechanics by explicitly *assuming* that such hydrogen sequences would be random in the sense of answering to no finite description.26)

But there is a big catch. Unless someone *looks* at a hydrogen atom, it need not have definitely emitted a photon or not. It can be, according to orthodox quantum mechanics, in what is called a *mixed state.* That is, unless it is being subjected to external measurements, the hydrogen atom can be in a state where it emits a photon with sixty percent probability and fails to emit a photon with forty percent probability, but does not unequivocally do either one!

The same sort of thing actually applies even to macroscopic systems such as the coin-flipper. The coin-flipper starts out in a certain state described by a certain wave function. The wave function evolves deterministically according to Schrödingers wave equation as time goes by. Unless someone is there to keep an eye on the flipper, it will soon enter a state of always getting tosses that are fifty percent heads and fifty percent tails. That is, *each* toss is 50/50.

It is difficult to assign any meaning to a statement such as this. How can you toss a coin, have it land (not on an edge) and be fifty percent tail and fifty percent head? *You* can’t … because you can only see one universe. If you could somehow split into two distinct people in two distinct universes, then you could see the coin come up heads *and* come up tails on the “same” toss.

One way of interpreting quantum mechanics is to claim that the universe actually does split like this everytime a decision has to be made. The idea is that if there is no sufficient reason for the world to choose *H* over *T* or *T* over H, then it chooses both.27

As the universe continues its dividing and redividing, every node in an infinite binary tree of possibilities is filled by a possible world. After ten coin flips, there will be coin flippers in 210 = 1024 different universes, one for each possible sequence of ten zeros and ones.

Curiously, such a branching universe contains less information than a universe that does not branch. For if there is only one coin-flipper, then a unique sequence of zeros and ones is generated, and this sequence is, in all likelihood, a random real coding up an irreducibly infinite amount of information. But if a coin-flipper splits in two every time the coin is tossed, then there is no one flipper you can point to in order to pick out a path through the binary tree.

If you start up a coin-flipper in a branching universe, then all you get is the occupation of every node of an infinite binary tree. This sort of setup is completely described by a finite number of words: “Take every possible finite sequence of zeros and ones.” If the universe does *not* branch, then the only way to describe what this particular flipper does is to state the actual sequence it generates: “.011010001010000110 …” In this case one can *hope* that there is some hidden order to the universe that finitely determines the sequence in question, but this hope may very well be unfounded.

This is an interesting point, and bears a little more discussion. If every possible universe exists, then there is no need to account for the special peculiarities of *this* universe (e.g., the facts that there is an ant crawling up my screen right now, or that there are 79 clover blooms in my backyard, or that sentient beings exist in this universe, or that space has three dimensions). If every possible universe exists, then there is no need to explain any peculiarity. Why is there an ant on my screen? No reason—there is another universe exactly the same except with no ant.

This situation is a bit like the Total Library. Given that every possible book is there, it would not really make sense to ask what the printer had in mind when he filled some given volume with “yam’s.” He did so because he had to do every possible book. In a sense, the Total Library contains no information at all!

So if every possible universe exists, then the cosmos can be rather simply specified by the injunction: “Take every possible universe.” Actually, a bit more than that would be needed. In particular, one would have to decide what constitutes a “possible universe.” A fairly conservative answer would be to take all the possible ways of filling a four-dimensional space-time with mass and energy consistent with Maxwell’s equations and Einstein’s field equations. And what about universes where the usual laws of physics do not hold, or universes where even the usual rules of logic do not apply? But it would take us too far afield to consider questions such as these.

It is very hard to believe seriously that every possible universe exists. But let us give it a try. If all possible worlds are out there, then every time I get in my car there is a world where I suffer a fatal accident. How, one is tempted to ask, do you stay in the worlds where you don’t get killed? The answer seems to be : You don’t. You are equally in all of those worlds. Really to accept this fact is a source, I would say, of profound liberation. Once you’re born, the worst has already happened to you.

The main unanswered question about the Many Universes model is how it is that one *seems* to oneself to be just in one universe. If everything is possible, why can’t you be aware of it? Maybe you can, or are all the time. If you examine your preverbalized thought patterns, they are rather different from ordinary consensus reality.

For example, if I am handed a bowl of walnuts, it may look in an instant like a cave, an old man’s face, clouds, the Matterhorn, a cat’s eye, hands, or *The Wreck of the Medusa.* When I see something for the first time, before I have decided what is is, it is many things at once. “Ah, yes, a bowl of walnuts,” I say shortly, and then I have only the one reality in my consciousness. But until I *name* the object and make it be one thing or another, I am in a mixed state—in many worlds.

Dreams are perhaps jumbled perceptions of many possible worlds. Language and ordinary thought form a sort of touchstone which keeps bringing you back to the same reality. You certainly do leave ordinary reality every time you fall asleep. If you woke up alone and with no memories but your dreams, could the dreams take over?

Despite all that has just been said, I do not believe very strongly in the full Many Universes theory. The universe we live in is so artfully constructed, so full of checks and balances, causes and coincidences, that it is hard to believe that this universe is just the product of a zillion coin-flips. There is so much holistic order in the universe that it seems implausible to suppose that we are just in some random one of all the possible worlds. There may be other universes, and we may be able to sense them in some manner, but I would expect each universe to have a certain overall patterning or essence.

This underlying pattern is what is sought when one looks for the shortest, most efficient complete description of a universe. But if the universe is indeed infinite, there seems to be no pressing reason why the underlying pattern of the universe should not be infinite as well. This brings us to Case 3.

This case, where one has an infinite universe with no finite description, actually splits into a number of subcases according to which level of infinity is assigned to the universe and to its descriptions. One could have a countable universe with a countable description, an uncountable universe with a countable description, or an uncountable universe with only an uncountable description. But it would be too confusing to consider these subcases here. Instead, let us further discuss the distinction between Case 2 and Case 3, looking at some concrete examples of how one might go about trying to code up the whole universe.

The most naive approach is to imagine that the universe is made of some finite or countably infinite collection of particles *m*_{0}, *m*_{1}, *m*_{2}, …, and that at each time *t*, each of these particles is of a definite type (electron, quark, photon, etc.) and has a definite position, orientation, and momentum. It is evident that all this information constitutes a set of real numbers that is at most countably infinite. Using the shuffling technique of the last subsection, all of these real numbers can be combined to form a single real number *U*(*t*), which represents an exhaustive description of the state of the universe at time *t*.

We who have heard of quantum mechanics all our lives must question whether any such number *U*(*t*) really exists, particularly since we know that we could never measure the position of anything with infinite precision, and especially not if the momentum is to be found with infinite precision as well. But for the physical determinists of the nineteenth century this objection had no force. They assumed that even if we could not measure it, *U*(*t*) existed at each *t*. The determinists also believed that it was not necessary to know all of the *U*(*t*). Indeed, they assumed that given *U*(*t*_{0}) at one time *t*_{0}, all of the remaining *U*(*t*) could be calculated on the basis of Newton’s Laws of Motion. So for an old-fashioned determinist, a complete description of the universe can take the form of a single real number *U*.

Simply by looking at the leaves on a tree, the wrinkles on one’s palm, or a sky full of clouds, one can quickly convince oneself that if such a universal number *U* has any finite description, then the length of this description must be very long. I would guess that the shortest natural number coding up a description of how to generate *U* would have to be longer than the number called *u*_{0} in “The Berry Paradox.”

An interesting side issue arises here. Even if some perfectly accurate description *U* of the universe exists, it seems likely that if we represented *U* by the clumsy expedient of putting numbers in books, then this representation of *U* would not fit in the universe. Of course, the most efficient representation of *U* is the universe itself, so at least one representation of *U* exists. But could we ever hope to have a desk-top or pocket-sized model of the universe? Only if matter is indefinitely divisible.

For if there is some smallest size particle, then any object in the universe will have less particles than the universe, and thus cannot serve as a scale model. But if there is no smallest particle size, then any portion of matter contains the same infinitely many particles, so it would be possible for some small region of matter to look exactly like the entire universe.

Let us describe a way of modelling this possibility. Consider the pattern in Figure 67, colloquially known as a *Phoebe Snow* by crapshooters. We can use this pattern as the base for an endless sequence of more and more detailed patterns of the same size. We do this by replacing each of the dots by a tiny Phoebe Snow to get a new pattern, replacing each dot in the new pattern by a tinier Phoebe Snow, and so on.

In Figure 69 I have reproduced plate 115 from Benoit Mandelbrot’s *Fractals*. This shows an enlarged picture of the fourth stage of the process just described. If the process is carried out an actually infinite number of times, we get something called a Fournier fractal (after E. E. Fournier D’Albe, who devised a similar pattern in 1907 to describe how he thought the galaxies might be arranged in space).

What we obtain is a universe that seems, at a glance, to be made of five chunks of matter in the Phoebe Snow arrangement. Upon closer inspection, each of these chunks resolves itself into five smaller subchunks, each of which upon closer inspection reveals itself to be a Phoebe Snow of five subsubchunks, etc. In other words, this fractal consists of five groups consisting of five groups consisting of five groups…. The phrase “five groups consisting of” is repeated ω times. Now, if a few of the initial repetitions of this phrase are left off, nothing is changed, so it is evident that each of the groups, subgroups, etc., has the exact same internal structure as the entire Fournier fractal universe.

I might add here that this could all have been carried out three-dimensionally by adding two dots to the basic pattern—one dot directly above the central dot, and one dot directly below the central dot, obtaining a “centered octohedron.” The final fractal would be obtained by endlessly replacing dots by smaller centered octohedra.

The point is that if matter is infinitely divisible, then it is conceptually possible that there are particles that are exact replicas of the whole universe. If one chooses to *identify* these particles with the entire universe, then the circular scale situation of the subsection, “Infinities in the Small,” arises. Note also that even if there is no circling back, such a Fournier fractal universe would contain only a *countable* infinity of groups, subgroups, etc. One unnatural feature of this model is that it is finite and *bounded*, in the sense of having outer boundary groups around an inner group. If we try to remedy this by curving the space into a hypersphere, then it is hard to see how the groups in the space can also be curved into hyperspheres, since the space itself normally has a dimension one lower than the space it is curved in. One way out of this problem would be to work with space of infinitely regressing dimensionality, so that one works with a hyperhyper … space, but this notion has never been studied.

In any case we were talking about ways of coding up the universe and the question of whether such codes are finite or infinite. The digression that just ended had to do with the question of whether it was, in principle, possible for the universe to include a scale model of itself.

We have already examined the physical determinists’ method of coding up the universe by a single real number *U.* Even the non-deterministic theories of physics would generally seem to allow for the universe’s instantaneous state being coded up by a real number, and for its entire history being coded up by a countable set of real numbers (give the state at each rational time coordinate), which can in turn be folded into a single universal real *U*, which may or may not be coded up by some magically efficacious finite formula.

There is something a bit unnatural in trying to describe the universe in terms of elementary particles, particularly in light of the fact that our knowledge of these particles is so derivative and uncertain. There are a number of facts that suggest the view that the data about these particles that we obtain are, in some sense, mind-produced. Perhaps it would be better to base our description of the world on actual human thought and experience.

That is, we would want to compile a list of all the things about the world that a person could find out. I might start, for instance, by mentioning the pen lying next to my typewriter. By way of describing it, I mention that it is black and shiny, but now I must pause to explain what “black” means. One component of what “black” means is certainly its association with the idea of “night.” Fully to express what “night” is, I must give specific examples of various nights I have experienced, for instance, the night in 1966 when Sylvia and I stood on a fragrant terrace looking out at the path the moonlight made on the Mediterranean. But really to describe how it felt to be there, I had better tell you about Sylvia. She was born in Budapest—here is a map of the city. A map is a sort of diagram drawn with a pen. Pens? They’re things you write with—I have one right here next to my typewriter; it’s black and shiny….

The fact is that when we try to describe any one object or experience fully in terms of other objects and experiences, more and more things get dragged in, including repeated appearances of the object originally being discussed.

There is no real contradiction or regress here, but it does seem that the body of possible experience is something like a neglected dish of hard candy; we try to pick one piece of candy up and the whole dishful comes along. It is probably impossible to describe any one thing in the world exhaustively without mentioning everything else as well. No matter what you start with you’re going to end up by mentioning the scar on my right index finger, the shape of the first sunspot to appear in 1292 B.C., the genetic makeup of the spirochetes that attacked Ivan the Terrible, and the nature of the galactic civilization that has evolved in the Whirlpool Nebula.

How can all this diversity be grasped as a mathematical unity? One approach would be to let The Description be the set of books in the Total Library that consist of true English descriptions of some aspect of the universe. Wittgenstein says, “The world is everything that is the case,” and we will let The Description be all the English descriptions of things that are indeed the case. The Description constitutes a set of books, which can be viewed as a set of code numbers. So The Description itself can be viewed as an infinite set of natural numbers, which can be coded, if desired, as a single real number that could be called *D*.

The restriction to English poses no real problems, since The Description will include an explanation of what each word means in terms of all the other words. Insofar as the attainable physical knowledge of the world constitutes only a part of “what is the case,” *D* really has more information than *U.*

Again, we might ask if *D* could possibly have a finite description of some kind. For instance, what if by some miracle, *D* turned out to be one over pi! People who believe that there is some ultimate answer to “it all” are in the position of hoping for just such a miracle. But anyone who has ever savored the endless diversity of nature must feel–and even hope–that the universe can never be fully captured by any finite schema, and that the pattern of the universe is, in a formal sense, random and unnameable.

The bounded and finite One of Plato and Parmenides seems no more worthy of worship than an overgrown computer. There is a harrowing passage in *Moby Dick* where Ahab stands on the deck in a thunderstorm, holding a lightning rod and ranting to his shipmates’ nameable God:

“There is some unsuffusing thing beyond thee, thou clear spirit, to whom all thy eternity is but time, all thy creativeness mechanical. Through thee, thy flaming self, my scorched eyes do dimly see it.”28

### WHAT IS TRUTH?

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The golden age of Greek philosophy was not very far in the past at the time of Christ. One of the very few representatives of the Greco-Roman world who appears in the Gospels is Pontius Pilate. In view of this fact, the following passage from the Gospel according to John takes on a certain significance as an archetypal confrontation between mystical and rational ways of thinking:

Pilate said to him, “So you are a king?” Jesus answered, “You say that I am a king. For this I was born, and for this I have come into the world, to bear witness to the truth. Every one who is of the truth hears my voice.” Pilate said to him, ‘What is truth?’ ” —John 18 (37-38).

The notion of truth leads to a number of logical difficulties. One of the most prominent of these difficulties is the Liar paradox, also known as the Epimenides paradox. Epimenides lived in Cnossus, the capital city of Crete, sometime before the time of Christ. It is generally believed that St. Paul is referring to Epimenides in the following passage: “One of themselves, a prophet of their own, said, ‘Cretans are always liars, evil beasts, lazy gluttons.’ This testimony is true.” —Epistle to Titus 1 (12-13).29

The paradoxical aspect of Epimenides the Cretan saying, “Cretans are always liars,” is that if what he says is true, then he must be lying. In view of this, it is especially ironic that Paul adds, “This testimony is true!”

The Liar paradox can be sharpened by considering the sentence: *A*) THIS SENTENCE IS FALSE.

That is, *A* is a sentence saying that *A* is false. If *A* is true, then what it says is true, so *A* is false. If *A* is false, then it is true to say that *A* is false, so *A* is true.

This is certainly a deplorable state of affairs. One way in which people have tried to get out of the paradox is to deny that A is either true or false—to assert that A simply happens to be a sentence that has no definite truth value. Now, we are certainly familiar with sentences that seem to be meaningless, rather than strictly true or false. “Virtue is triangular” and “Donald Duck weighs 62.8 pounds” are sentences that we would hesitate to call definitely true or definitely false. So why couldn’t A be another such sentence?

This exit route can be closed in the following way. Consider the sentence: *B*) THIS SENTENCE IS NOT TRUE.

*B* is a sentence saying that *B* is not true. Now, every sentence is either true or not true. If *B* is true, then *B* is not true. If *B* is not true, then *B* is true. Therefore, *B* is both true and not true—which is a contradiction.

It certainly seems that any sentence must either be true or not true, where “not true” is taken in the broadest possible sense of “false, meaningless, contradictory, or impossible to verify.” So it seems dishonest to try to escape the present form of the Liar paradox by denying that every sentence is either true or not true.

We might, instead, try denying that *B* is a sentence. There is certainly something peculiar about *B.* It refers to itself with the phrase THIS SENTENCE, and if we replace THIS SENTENCE by a quotation of the sentence in question, namely by “This sentence is not true,” then we are no better off than before. Repeating this substitution leads to an infinite regress:

THIS SENTENCE IS NOT TRUE.

“THIS SENTENCE IS NOT TRUE” IS NOT TRUE.

““THIS SENTENCE IS NOT TRUE” IS NOT TRUE” IS NOT TRUE …

.

.

.

“ “ “…” IS NOT TRUE” IS NOT TRUE” IS NOT TRUE.

Note that the limiting sentence at the bottom consists of an ω-sequence of “ ‘s running from left to right, followed by an ω-sequence of” IS NOT TRUE’s running from right to left.

There is actually nothing inherently bad about infinite regresses. Josiah Royce has made the point that an infinite regress can usually be avoided by looking at the situation in a schematic way. For instance, if I say that there is a sentence *B* that says *B* IS NOT TRUE, then there is no infinity present. It is only when I insist on eliminating the symbol “*B*” that the infinite regress arises.

Recall that we say a very similar situation in the “Infinities in the Mindscape” section, when we discussed a mind that consists of pure self-awareness. This was modelled as a set *M* whose only member is *M.* It is possible to grasp the essence of the set *M* immediately and all at once, but if we try to eliminate the symbol “*M*”, we get the infinitely regressing definition of *M* as {{{… …}}}.

The traditional belief was that if a line of thought leads to an infinite regress, then this line of thought is invalid.30 This belief is founded on the notion that infinity is inherently contradictory and even incoherent. But Cantor has delivered us from this superstitious fear of the infinite. And not a moment too soon, for in 1893 F. H. Bradley published a book, *Appearance and Reality*, which seems to show that just about *any* sentence leads to an infinite regress when thoroughly analyzed.31

Bradley’s argument goes something like this. Ordinarily we think of the world as being made up of various individuals *a, b*, …, which stand in various relations *R, P*, … to each other. For instance, for me to say that this *X* is to the left of *this X* is to say that a certain relation *L* (“ to the left of”) is satisfied by the object *a* (the first *X* mark) and the object *b* (the second *X* mark) taken in that order. This is abbreviated as *L*(*a, b*).

Now, Bradley continues, we can think of relations as themselves being higher-order objects that can, in turn, stand in various higher-order relations to other relations and objects. We do this, for instance, when we say, “It is more blessed to give then to receive.” Returning to the example above, we can view the relation *L* as a (higher-order) object and say that *L(a, b*) really means that *L, a*, and *b* have a certain higher-order relation-call it satisfaction- to each other. This is, we can say that *L(a, b)* really means *S(L, a, b*) where *S* is the “first-order satisfaction relation” such that in general *S(R, x, y)* holds if and only if *x* stands in the relation *R* to *y.*

But now we can introduce a still higher-order relation *S*′ and say that *S(L, a, b)* really means *S*′(*S, L, a, b*) where *S*′ is the “second-order satisfaction relation” such that in general *S*′(*Q, R, x, y*) holds if and only if *Q*(*R, x, y*). Indeed, Bradley argues that not only is it *possible* to start down this path, but that it is *necessary.* For he feels that to assert a statement such as *L*(*a, b*) is to say that the relation *L*, the object *a*, and the object *b* stand in a certain relation to each other.

And, of course, there is no stopping, and we have the infinite regress:

*L*(*a, b*)

*S*(*L, a, b*)

*S*′(*S, L, a, b*)

*S*′′(*S*′, *S, L, a, b*)

.

.

.

… *S*′′, *S*′, *S, L, a, b*).

The moral is that even a statement as simple and unproblematic as “this *X* is to the left of *this X*” can lead to an infinite regress, so the mere fact that the Liar paradox sentence *B* leads to an infinite regress does not automatically mean that *B* is not a sentence and that *B* is, therefore, exempt from having to be either true or not true. In other words, we should not dismiss the Liar paradox out of hand simply because it leads to an infinite regress—for Bradley has shown tha*t every* sentence leads to an infinite regress.

Before going any further, let’s eliminate the symbolism and try to see what Bradley has really done. His basic intuition seems to be that nothing can be linked to anything else without a mediating relation. Thus, *L* links *a* to *b, S* links *a* to *L* and *L* to *S*′ links *a* to *S* and *S* to *L* and *L* to *b*, and so on.

We can visualize this geometrically by starting with an *a* and *b* with a sort of chasm in between them. Now, the relation *L* can be viewed as a bridge across the chasm. Continuing, we can think of *S* as a pair of bridges spanning the gulfs between *a* and *L* and *L* and *L* and of *S*′ as a set of four bridges connecting *a* to *S, S* to *L, L* to *S*, and *S* to *b.* In the limit we get a fractal, which is partially drawn in Figure 73.

Leaving aside, for now, Bradley’s concern with how it is that disparate language elements are unified into a sentence, let us consider what the sentences in his infinite regress actually *say.*

*L(a, b)* says *a* is to the left of *b.*

*S(L, a, b)* says “*a* is to the left of *b*” is true.

*S′ (S, L, a, b)* says “ “*a* is to the left of *b*” is true” is true.

… *S ″, S′, S, L, a, b)* says … “ “ “*a* is to the left of *b*” is true” is true” is true”….

Another way of putting it is that Bradley believes that truly to assert some sentence *E*, one must really assert each of the following infinite regress of sentences: *E*, “*E*” is true, “*E*” is true” is true, … This regress tends toward the limit … “ “ “*E*” is true” is true” …, where on the left we have an ω-sequence of left-hand quotation marks running from right to left, and on the right we have an ω-sequence of *is true*” ‘s running from left to right.

Can we avoid this infinite regress by insisting that once you have said *E*, it adds nothing also to say “*E*” *is true*? This is the case for certain artificial sentences. We can construct such an artificial sentence like this: *C)* THIS SENTENCE IS TRUE.

*C* says that *C* is true, so *C* and “C” *is true* are the same sentence. If we analyze *C)* as we did *B*), we see that *C)* actually has the form: “““… …” IS TRUE” IS TRUE” IS TRUE, where an ω-sequence of quotes moves in from the left, and an ω-sequence of” IS TRUE’s moves in from the right. Clearly, tacking on an extra quotation mark in front and an” IS TRUE in back changes nothing. So for the artificial sentence *C, C* and “*C” is true* are the same. Incidentally, you might wonder whether THIS SENTENCE IS TRUE is true or not. I don’t know. All I can tell you offhand is that if it is true, then it is true; and if it is false, then it is false … which is not too helpful.

Returning to the main line of thought, I would like to stress the fact that for most sentences *E, “E” is true* really is a sentence that is different from *E*. To understand a sentence such as “*a* is to the left of *b*,” we need only know where *a* and *b* are and what “left” means. But to understand “It is true that *a* is to the left of *b*”, we have to know what “true” means as well. And this is not so easy to know.

In point of fact, there can be no finite complete description of truth. *Truth is undefinable.* As we will see, this will be our way out of the Liar paradox. For if truth is undefinable, then the word “truth” cannot properly be used to stand for the full concept of truth, so the Liar sentence *B* really does not mean what we thought it did, and the paradox is avoided.

But before thinking that through, let’s look at Alfred Tarski’s 1934 proof that truth is not definable.32 Consider the Total Library. Some of the “books” will assert sentences, and some of these sentences will be what we think of as *true* sentences. If, for instance, the text of *B _{5389}* is simply “Snow is white,” then we would say that

*B*

_{5389}

*is a true book*, or, abbreviating,

*T*(

*B*

_{5389}).

In general, *T(B*_{n}) if *B _{n}* is a book that is true. Recall that in the last subsection we referred to the set of all true books as The Description. You might ask who is going to decide which of the

*B*are true—but let’s dodge that for now by saying we’ll let God decide which books are true.

_{n}So *T(B _{n})* holds if and only if

*B*is true. Now I wish to prove that there can be no finite description of the

_{n}*T*predicate. There can be no finite accounting for which books God will say are true.

For suppose that there were a finite blueprint or program for building a “truth machine” such as the one illustrated in Figure 74. The truth machine has a slot where you can put a book *B _{n}* in. The truth machine scans the pages of

*B*

_{n}, and if

*B*is true, then it is placed in a nice orderly row with all the other true books. But if

_{n}*B*is

_{n}*not*true, then the truth machine spits it out onto the scrap heap of history.

O.K. So, now have a finite set of instructions for building a truth machine. Let *K* be the book that goes as follows:

“Imagine building a truth machine according to the following set of instructions. [Give the instructions.] Now, the assertion that this book wishes to make is the following: The truth machine will not say that this book is true.”

Toss *K* into the truth machine’s mouth and watch the fun begin!

Fun, that is, if you enjoy seeing valuable machinery destroy itself. The truth machine can’t say that *K* is false, since then *K* would be true—in that it predicts that the truth machine will not say that *K* is true. And the truth machine can’t say that *K* is true, since if the machine says *K* is true, then *K* is false—since it predicts that the machine will not say *K* is true. So *K* sticks in the truth machine’s craw: The machine can’t say YES and it can’t say NO.

So, in fact, the truth machine never will say that *K* is true. But that’s what *K* says. We who look in from the outside can see that *K* really is true, in the absolute and undefinable sense of the word “true.” *K* is a true book, but the truth machine is unable to recognize this fact!

Speaking a bit more formally, we have proved that for any finite set of instructions *S* for building a truth machine *M _{S}*, there is a finite book

*K*such that

_{S}*K*is true and

_{S}*M*

_{S}cannot recognize this fact.

*K*is, as above, the book saying, “M

_{S}_{S}will not say that

*K*is true,” and when we give

_{S}*K*to

_{S}*M*goes into an endlessly regressing loop and never says anything again.

_{S}, M_{S}There can be no finite description of a truth machine that singles out as true all the true books. There is no robot we could build to winnow The Description out of the Total Library. Truth is undefinable.

The solution to the Liar paradox is now at hand. In itself, the sentence: *B)* THIS SENTENCE IS NOT TRUE is not really a meaningful utterance. Some description *S* of what is meant by TRUE must be appended, so that *B* really becomes *K _{S} : K_{S})* THIS SENTENCE IS NOT TRUE ACCORDING TO DESCRIPTION S OF TRUTH.

In terms of description *S, K _{S}* will be neither true nor untrue, for

*S*cannot reach a decision on

*K*

_{S}. So there is no paradox.

In terms of our absolute, but unformalizable, notion of truth, *K _{S} is* true. Better and better descriptions of truth can be obtained, but no finite description can ever exhaust the unnameable concept that we point at with the symbols T-R-U-T-H.

A truth machine description *S* cannot always correctly decide the truth of statements mentioning *S.* But it is possible, given *S*, to come up with a better truth machine description *S*′ that does everything *S* did and also is able to correctly decide the truth of statements about *S*. (*S*′ could, for instance, include all the rules of *S*, plus the new rule, “*K _{S}* is true.”)

This move can be used to generate an infinite sequence *S _{n}* of truth machine descriptions. We let

*S*be some truth machine description that decides about sentences that do not mention truth at all, and for each

_{0}*n*+ 1 we let

*S*

_{n=1}=

*S*′. That is

_{n}*S*

_{n+1}decides everything

*S*did, and decides sentences about

_{n}*S*as well.

_{n}With this in mind, we can arrive at a better understanding of what underlies Bradley’s regress. Suppose that you want to state some declaration *E* in such a way that it is absolutely clear that *E* is true.

If you simply say “*E*,” then people may think that you are only reciting poetry or framing a hypothesis, so you want to say something like “*E* is true.” Unfortunately, the people you are talking to are positivists, and they insist that you define every term that you use—and it is impossible to define the absolute notion of truth that you had in mind. So, you describe some definition *S*_{0} of truth and say, “*E* is true according to *S*_{0}.” But now you feel that you should stress that this last sentence is true, and the regress begins:

*E*.

“*E*” is true according to *S*_{0}.

“ “*E*” is true according to *S*_{0}” is true according to *S*_{1}.

“ “ “*E*” is true according to *S*_{0}” is true according to *S*_{1}” is true according to *S*_{2}.

.

.

.

No final satisfaction is ever reached. Indeed, this regress can be carried on past ω. For if *S*_{ω} is the truth description that feeds a given book into one after another of the *S _{n}* until an answer appears, then

*S*

_{ω}is still finitely described and we can move on to

*S*

_{ω+1}=

*S*

_{ω}′.

Of course, our absolute notion of truth *should* be such that if *E* is true, then “*E* is true” is true, and so on. But this absolute notion is not finitely describable. If we restrict ourselves to finitely describable notions of truth, then Bradley’s regress is unavoidable.

It is, of course, possible to deny that any absolute notion of truth exists, and to insist that all there will ever be is better truth definitions, tending toward a limit that is wholly imaginary. This point of view is analogous to the point of view that admits that there are arbitrarily large natural numbers, but denies that the actually infinite limit ω of this sequence exists. Even more analogous is the viewpoint of those who admit the existence of the various infinite sets, but deny the existence of the unified existence of all sets at once in Cantor’s Absolute. These are all variations of the One/Many problem that will be discussed in Chapter 5.

The critical reader will have noticed certain weak points in the proof that truth is undefinable. In particular, he may wonder how one can legitimately construct a book such as *K _{S}* that refers to itself without using the somehow transcendental phrasing, “this book,” and he may wonder if the proof is not somehow circular or fallacious, since it seems to mention the very notion of truth whose undefinability is being proved. In the next chapter (and in Excursion II) I show how a version of this proof can be constructed so as to be completely unimpeachable.

To sum up, we have shown that truth is a concept that cannot be rigorously defined in a finite way. Because of this, it is possible to assert that THIS SENTENCE IS NOT TRUE is not really a sentence, and is thus exempt from having to be either true or not true. This is not an entirely satisfactory solution, since we feel that a collection of words that is neither true nor not true should *really* be called not true. Like all good paradoxes, the Liar paradox resists any final resolution and endures as “an eternal crevice of unreason.”

### CONCLUSION

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The three columns of the following table summarize the three sections of this chapter. In each case one starts with a familiar infinite concept a), and runs into a paradox, b). These three paradoxes have a certain similarity in that each of them has to do with *semantics*, that is, each paradox centers on the process of determining the *meaning* of strings of symbols. One way of resolving these paradoxes is to insist that the key word (“name,” “nameable,” “true”) points toward, but cannot actually name or define the needed concept with the requisite precision, and then to assert that the statements in row b) can be regarded as meaningless. But one cannot just stop here. It is not satisfying to “solve” paradoxes by dismissing them as meaningless. I am reminded here of the master puzzlist Sam Loyd’s comments on Alexander the Great’s method of undoing the Gordian knot:

Alexander the Great, it is said, made many ineffectual attempts to untie some of the knots, but finally becoming enraged at his want of success, drew his sword and cut the cord, exclaiming that ‘such is the common sense way to get a thing when you want it.’ Strange that those familiar with the story and its contemptible climax endorse it with a certain air of assumed pride when they have surmounted some difficulty and exclaim: ‘I have cut the Gordian knot!’33

In row c) we have taken each of the paradoxes and replaced the nebulous key words by precise approximations (“*M*_{1}-name,” “*M*_{2}-nameable,” “true according to *M*_{3}”). The *M*_{i} are intended to be any finitely describable system, e.g., an appropriately programmed digital computer, or perhaps even a human being (for it may be that a biological organism does admit of an exhaustive finite description). It is to be understood that *M _{1}* is arranged so that a given finite input

*B*causes

*M*

_{1}to print out some finite natural number

*n*; that

_{B}*M*

_{2}is set up so that any given finite input

*B*converts

*M*

_{2}into a device that will give out a digit

*r*for each requested

_{B · n}*n*; and that

*M*takes any given finite input

_{3}*B*and types out “true” or “not true.” The key fact is that none of the

*M*always work properly, that there are certain

_{i}*B*’s that cause these

*M*’s to run forever without ever outputting anything. In particular, each of the three paragraphs in row c) will cause the relevant

_{i}*M*to enter an endless loop.

_{i}a) The natural numbers. | The real numbers. | Truth. |

b) Let the Berry number be the first natural number which has no name as short as this sentence. | Let the Richard number be the real number obtained by diagonalizing the list of all nameable reals. | This statement is not true. |

c) Let M be the following system: [Description of _{1}M_{1}]. Print out the first number which has no Mi-name as short as this. |
Let M be the following system: [Description of _{2}M Generate the real number obtained by diagonalizing over all reals which _{2}].M_{2} names. |
Let M be the following system: [Description of _{3}M_{3}]. M will not say that this paragraph is true._{3} |

d) There is some number u such that M is unable to generate any natural numbers with complexity greater than _{1}u. |
There is a nameable real number which M cannot name._{2} |
There is a true sentence which M cannot recognize as true._{3} |

e) No finite system can generate arbitrarily complex patterns. | No finite system can understand everything. | No finite system can define truth. |

Because of this we can draw the conclusions in row d). Since *M*_{1} cannot find the natural number described in row c), it must be that cannot locate *any* number whose shortest description is longer than the length *u* of that paragraph in row c). For any input *B*, either *M*_{1} gives out a number whose complexity (shortest description length) is less than *u*, or *M*_{1} runs forever. In running forever it will go *past* numbers (of seconds) of arbitrarily large complexity, but it is unable to stop and point out any of these numbers with complexity greater than *u*. Since this argument applies to any finite system *M _{1}*, we can draw the general conclusion in row e).

Since *M _{2}* does admit of a precise finite description, the description in row c) does actually describe a specific real number. It is just that this description cannot be unraveled by

*M*

_{2}. In this sense there is some meaningful paragraph that

*M*

_{2}does not understand, and the conclusion in row e) can be drawn. Recall from the section on Richard’s paradox that analyzing the construction of the

*M*

_{2}-Richard number shows that

*M*

_{2}is also unable to name the real number

*T*

_{M2}that codes up its own translation process. So we might improve the conclusion in row e) to something like; no finite system can finitely describe the process by which it converts words into thoughts.

Since *M*_{3} never draws a conclusion on the paragraph in row c), we know that *M*_{3} will not say that that paragraph is true. So it is in fact true, although *M*_{3} cannot recognize this fact. Thus, we reach the conclusion in row e), which can also be phrased this way: for any given finite system there is a truth that the finite system in question cannot recognize as true.

This last conclusion is an imprecise statement of Gödel’s First Incompleteness Theorem, which will be considered in detail in the next chapter, where we will also attempt to use these facts to draw some conclusions about the nature of human and mechanical minds.

So we have learned that for any finitely give system *M*, there will be a number of things that *M* cannot name, describe, conceive of, understand. For any finite system *M*, there are things that are *unnameable relative to M*. Now we must ask if there is anything that is *absolutely unnameable*, that is, beyond the comprehension of any finite system whatsoever.

In Chapter 1 we considered the related question of whether anything is actually (as opposed to potentially) infinite. We knew that for any given natural number *n*, there is a larger natural number (e.g., *n* + 1). The question was whether there is any number such as ω that is bigger than every natural number at once.

I discussed the possibility of physical and mental actual infinities the “Physical Infinities” and “Infinities in the Mindscape” sections; in “The Absolute Infinite” I introduced the Absolute as something that is assuredly infinite, if indeed it exists.

In the “Coding the World” subsection in this chapter the issue of whether any physical and absolutely unnameable reals exist was handled; it should be evident from our previous discussions of the Absolute that it is absolutely unnameable. However, we have not yet said much on the question of whether any absolutely unnameable mental objects can be found.

We have already seen that the set of all true books is an unnameable mental object. But this set is rather nebulous, and one might question if it really exists. The same would apply to the unnameable process by which we translate words into thoughts. In the next chapter we will see that there is a fairly definite mental object that cannot be completely specified in any finite way. This is the set of all true statements about natural numbers.

### PUZZLES AND PARADOXES

*(Answers, p. 299)*

1. Show that ten tetrated to the four is greater than googolplex.

2. A Berry-paradox-style argument is sometimes given to “prove” that every natural number is interesting. Try to construct such an argument.

3. Find the lengths of the hypotenuses in this spiral arrangement of right triangles:

4. Suppose that I find in the Total Library a book that I believe to be a correct, detailed description of your entire life, past and future. You do not like this, as you feel your future is, and should remain, undecided, if I show you the book in question, can you prove that I am wrong?

5. This is the classical paradox of the Crocodile: “A Crocodile had stolen a Baby off the banks of the Nile. The Mother implored him to restore her darling. ‘Well,’ said the Crocodile, ‘if you say truly what I shall do I will restore it: if not, I will devour it.’ “34 What should the mother say?

6. What is the meaning of the following sentence: “APPENDED TO ITS OWN QUOTATION IS FALSE” APPENDED TO ITS OWN QUOTATION IS FALSE.

7. John Barth begins his *Lost in the Funhouse* with the infinite story: ONCE UPON A TIME THERE WAS A STORY THAT BEGAN ONCE UPON A TIME THERE WAS A STORY THAT BEGAN ONCE UPON A TIME THERE WAS A STORY THAT BEGAN….36 This involves a single left-to-right ω-sequence. Can you think of an appropriate right-to-left ω-sequence to close off Barth’s story with?

8. Say that we have established some starting assumption *A*, and wish to draw a specific conclusion *C*. In order to do this, we normally proceed by proving the implication (if *A*, then *C*). But suppose that some stubborn individual denies that *C* follows inevitably from *A* and (if *A*, then *C*). Then we must establish not only *A* and (if *A*, then *C*), but also (if *A* and (if *A*, then *C*), then *C*). Show that if our interlocuter remains stubborn enough, he can in this way force us into an infinite regress.37

## CHAPTER FOUR

## ROBOTS AND SOULS

Are people just complicated robots … or do we have souls? On the one hand, being conscious certainly *feels* like something more than the mechanical working out of a computer program. But on the other hand, what could a soul possibly be? How could it act on matter? Might a machine have a soul?

A theorem of mathematical logic, Gödel’s Incompleteness Theorem, is widely thought to have a bearing on this constellation of problems. In this chapter we will begin by investigating Gödel’s famous theorem and end by making some speculations about mechanical intelligence and the nature of consciousness.

The first section gives a quick overview of Gödel and his Incompleteness Theorem. In the following section I describe a series of conversations that I had with Gödel about his theorem and some related matters. “Towards Robot Consciousness” contains a more detailed treatment of Gödel’s theorem and explains exactly what Gödel thought the consequences of his theorem were for the field of Artificial Intelligence. (An even more detailed treatment is to be found in Excursion II.) In “Beyond Mechanism,” I explore Gödel’s claim that there is a nonmaterial component to human consciousness.

### GÖDEL’S INCOMPLETENESS THEOREM

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In the summer of 1930, the twenty-four-year-old mathematician Kurt Gödel proved a strange theorem: *mathematics is open-ended.* There can never be a final, best system of mathematics. Every axiom-system for mathematics will eventually run into certain simple problems that it cannot solve at all. This is Gödel’s Incompleteness Theorem.

The implications of this epochal discovery are devastating. The thinkers of the Industrial Revolution liked to regard the universe as a vast preprogrammed machine. It was optimistically predicted that soon scientists would know all the rules, all the programs. But if Gödel’s Theorem tells us anything, it is this: Man will never know the final secret of the universe.

Of course, anyone can *say* that science does not have all the answers. What makes Gödel’s achievement so remarkable is that he could rigorously *prove* this, stating his proof in the utterly precise language of symbolic logic. To come up with a mathematical proof for the incompleteness of mathematics is a little like managing to stand on one’s own shoulders. How did Gödel come to think of such a proof? What kind of person was he?

Kurt Gödel was born on April 28, 1906, in Brünn (Brno), Czechoslovakia, at that time part of Austria-Hungary. His family was part of the sizable German minority in Brünn, and his father was the successful manager of one of the city’s numerous textile mills. Gödel’s childhood was marred by a case of rheumatic fever at the age of six. He recovered from the disease, but for the rest of his life he had morbid fears about his health.1

He attended the University of Vienna, starting as an undergraduate in 1923 and earning his doctorate in mathematics in 1930. Vienna was a terribly exciting place to be in those years. The beginnings of psychoanalysis, twelve-tone music, modern architecture, and non-representational painting can all be traced to this period—with Sigmund Freud, Arnold Schönberg, Adolf Loos, and Oskar Kokoschka all active in Vienna.

But, more importantly for Gödel, this was also a period of great philosophical ferment in Vienna. In 1921 the Viennese intellectual Ludwig Wittgenstein had published his gem-like *Tractatus Logico-Philosophicus.* And logical positivism was being founded and developed by a group of philosophers known as the “Vienna Circle.” Gödel’s principal teacher, Hans Hahn, was a prominent member of this group, as were Moritz Schlick, Philipp Frank, and Rudolf Carnap. The Vienna Circle held its meetings in a seminar room near the mathematics department, and Gödel attended these meetings regularly.

The basic credo of logical positivism is summed up by Rudolf Carnap’s manifesto: “We give no answers to philosophical questions and in deed *reject all philosophical questions*, whether of Metaphysics, Ethics or Epistemology.”2 The idea was that abstract philosophical statements such as, “All is One,” are meaningless—not true or false, but simply without content. This position was based on the so-called “Verifiability Principle,” according to which the *meaning* of a statement is equated with the *method for verifying* the statement. Since the positivists could see no way scientifically to document metaphysical statements like, “All is One,” or “The Absolute is outside time,” these statements were regarded as wholly devoid of significance.

This basically negative part of logical positivism was greatly influenced by Wittgenstein’s celebrated *Tractatus.* This short and aphoristic book offers the following solution to the problems of traditional philosophy: “What we cannot talk about we must pass over in silence.”3 But it should be pointed out that Wittgenstein, although friendly with members of the Vienna Circle, was never quite a logical positivist. On the contrary, he sometimes sounds like a Zen mystic. He elegantly describes his position in the *Tractatus:*

“We feel that even when *all possible* scientific questions have been answered, the problems of life remain completely untouched. Of course there are then no questions left, and this itself is the answer. The solution of the problem of life is seen in the vanishing of the problem.”4

The positive part of logical positivism was a program to unify all of science, using the language of symbolic logic. The inspiration came from Whitehead and Russell’s *Principia Mathematica* of 1910.5 In this monumental work (three mammoth volumes), it is shown how all of our familiar mathematical concepts and facts can be logically derived from certain very simple and primitive principles of reasoning. The logical positivists hoped to put other branches of science, including physics and psychology, on a similarly rigorous footing.

The principal achievement of Whitehead and Russell was to get a mechanically precise definition of what we mean when we say that some statement *follows logically* from some other statements. With this definition in hand, the “formalist” mathematician David Hilbert pointed out that mathematics was now only a matter of choosing the right axioms and examining the logical consequences of these axioms. The positivists hoped to extend this approach to all the sciences, and even to all of human thought.

In order to grasp the implications of such a programmatic approach to human knowledge, I would like to run a little thought-experiment here, an alternate-world fantasy. Imagine:

A complete axiom-system for mathematics was obtained by 1950. The system was called MT for Mathematical Truth. It was theoretically established that any true mathematical statement could be proved from MT, and that any false mathematical statement could be disproved from MT. Thus, the axioms of MT, along with the Whitehead-Russell-Hilbert rules for logical deduction, captured the whole of mathematics.

The existence of the complete theory MT did not really begin to affect mathematicians until the twenty-first century. They went on much as before, using their intuition and ingenuity to find ways of combining the various axioms of MT to yield logical proofs of interesting theorems. But in the year 2000, the computers became big enough to take over. In the space of ten years a single linked system of Josephson-junction processors had made mathematicians as obsolete as slide-rules. The system was called MTM for Mathematical Truth Machine.

The functioning of MTM was as follows. MTM was programmed with the basic axioms of the complete system MT. What MTM did was simple. It exhaustively worked out all the logical consequences of these axioms: first all the theorems with one-step proofs; then all the theorems with two-step proofs; then three; and, before long, three million; and so on and on.

As MTM proved theorem after theorem, these were added to its systematic master list. If you wanted to know about some mathematical problem (“Is Fermat’s Last Theorem true?” “What is the solution to this differential equation?” “What is the shortest route connecting these ten cities?”) you fed your question into MTM, and MTM would search its master list for your answer.

If the answer was already in the master list, fine. If not, you had to wait a bit, but sooner or later MTM would get to the theorem answering your question. There was no point consulting a mathematician instead, since MTM had already gone far beyond all logical derivations short enough to be’ humanly comprehensible.

All this was fine with everyone except the mathematicians. A few of them rebelled to create a new “surreal mathematics” based on deliberately false and inconsistent assumptions. But MTM one-upped them by taking a spare hour to work out the most interesting false theorems of “surreal mathematics.” With its ever-increasing store of mathematical and logical fact, MTM was getting faster and faster. You could feed it any set of axioms and it could work out the most interesting consequences in next to no time at all.

The physicists were the next to go the way of the mathematicians. Late in the 1990s an Israeli graduate student had achieved the final unification of General Relativity with Quantum Mechanics. A simple list of axioms, twenty-five in number, summarized all the laws of nature. This theory, called PT for Physical Truth, was programmed into a computer linked up with MTM. The new system, called PTM for Physical Truth Machine, began systematically working out the consequences of PT. Soon the three-body problem was solved, an explanation for the mass of the electron was produced, the exact age of the universe was calculated, and several methods for safe nuclear fusion were discovered.

A critical mass of knowledge had been reached by MTM and PTM. In the next few years, complete theories were found for biology, psychology, and sociology. A planet-wide system of linked computers combined all of these complete theories to produce the god-like STM, for Scientific Truth Machine.

Any scientific question at all was best answered by consulting STM. Either STM had already worked out the answer, or it soon would. No scientist could know as much as STM, so independent work was useless. During the Italian Renaissance there were certain mathematicians who made their living by being able to solve cubic equations. Imagine their dismay if one of the princes had been given the simple formula for solving cubics, along with a pocket calculator to work out the solutions! This was the situation in which the scientists now found themselves. The utterly mechanical working out of a complete theory had replaced any need for scientific intuition or creativity.

The last step came suddenly, in the year 2060. An embittered Argentine scientist had, with the help of the STM, worked out a complete theory of aesthetics. The immutable laws of what makes a great novel, painting, or symphony were incorporated into an axiom system called AT for Artistic Truth. The clandestinely assembled ATM sprang into action and began producing short works magnificently expressive of the condition of man in the cosmos.

Over the protests of the artists, the government coupled ATM with STM to get UTM, the Universal Truth Machine. There was no longer any need to do anything. Whatever any person wanted to know or do or say would, sooner or later, be done better by UTM. Terrorist acts against UTM were impossible, as UTM had a complete theory of human behavior, and could predict and ward off any such attack. About the only thing people still had left for their own was sports. A UTM terminal was placed in each home, and the world slid toward senescence, watching the tube.

Depressing? Don’t worry!

In 1930 Kurt Gödel proved that there can never be a UTM (Universal Truth Machine). There can’t even be an MTM (Mathematical Truth Machine). There *is* no complete set *MT* of axioms for mathematical truth. Any system of knowledge about the world is, and must remain, fundamentally incomplete, eternally subject to revision.

Of course, the future could follow a scenario *similar* to what I just outlined. The difference, thanks to Gödel, is that the machines will never have *all* the answers. There will always be room for a creative person who can think of a better way of doing things.

Try to catch the universe in a finite net of axioms and the universe will fight back. Reality is, on the deepest level, essentially infinite. No finitely programmed machine can ever exhaust the richness of the mental and physical world we inhabit.

The proof of Gödel’s Incompleteness Theorem is so simple, and so sneaky, that it is almost embarrassing to relate. His basic procedure is as follows:

1. Someone introduces Gödel to UTM, a machine that is supposed to be a Universal Truth Machine, capable of correctly answering any question at all.

2. Gödel asks for the program and circuit diagrams of the UTM. The program may be complicated, but it can only be finitely long. Call the program *P*(*UTM*) for Program of the Universal Truth Machine.

3. Smiling a little, Gödel writes out the following sentence: “The machine constructed on the basis of the program *P*(*UTM*) will never say that this sentence is true.” Call this sentence *G* for Gödel. *Note that G is equivalent to “UTM will never say G is true*.”

4. Now Gödel laughs his high laugh and asks UTM whether *G* is true or not.

5. If UTM says *G* is true, then “UTM will never say *G* is true” is false. If “UTM will never say *G* is true” is false, then *G* is false (since *G =* “UTM will never say *G* is true.”) So if UTM says that *G* is true, then *G* is in fact false, and UTM has made a false statement. So UTM will never say that *G* is true, since UTM makes only true statements.

6. We have established that UTM will never say *G* is true. So “UTM will never say *G* is true” is in fact a true sentence. So *G* is true (since *G* = “UTM will never say *G* is true.”).

7. “I know a truth that UTM can never utter,” Gödel says. “I know that *G* is true. UTM is not truly universal.”

Think about it—it grows on you.

The gimmick in Gödel’s proof is very similar to the gimmick in the famous Liar paradox of Epimenides: “I am lying,” says Epimenides. Is he? Or, as in “What is Truth,” define *B* to be the sentence “*B* is not true.” Is *B* true? The problem is that *B* is true if and only if *B* is not true. So *B* is somehow outside the scope of the applicability of the concepts “true” and “not true.” There is something viciously meaningless about the sentence *B*, and one is inclined just to try to forget about it.

But Gödel’s *G* sentences cannot be so lightly dismissed. With his great mathematical and logical genius, Gödel was able to find a way (for any given *P*(*UTM*) actually to write down a complicated polynomial equation that has a solution if and only if *G* is true. So *G* is not at all some vague or non-mathematical sentence. *G is a specific mathematical problem that we know the answer to, even though UTM does not!* So UTM does not, and cannot, embody a best and final theory of mathematics.

Gödel’s Incompleteness Theorem flew in the face of the formalist and logical positivist movements of the time. But everyone who was capable of following the many steps of the detailed proof was forced to concede its correctness. Gödel became famous.

When the Second World War broke out, Gödel moved to Princeton, New Jersey, to take up a permanent position at the Institute for Advanced Study. This famous “college without students” had been established a few years earlier with the financial help of Louis Bamberger, the department-store magnate. The aging Einstein was also there, and he became one of Gödel’s few intimates. They could often be seen walking the great lawn in front of the Institute’s Fuld Hall discussing relativity. Gödel himself published a curious paper in relativity, in which he described a special class of universe where time-travel by rocket ship is possible.6

Gödel did some of his most interesting work during the 1940s. Soon after arriving in the United States he published his only book, a monograph on set theory called *The Consistency of the Continuum Hypothesis*.7 This book contains his proof that Cantor’s Continuum Hypothesis cannot be disproved from the existing axioms of set theory. As with his Incompleteness Theorems, it was not only the statement of the theorem, but also the method of proof that had such an impact on mathematics and philosophy. Here Gödel found a whole new way of thinking about the class of all sets and discovered certain absolute features of the mathematical universe that had never been suspected.

In the mid-1940s he wrote a pair of rather philosophical papers aimed at the non-specialist, “Russell’s Mathematical Logic” and “What is Cantor’s Continuum Problem?”8 These beautifully written papers reveal Gödel as anything but a logical positivist. He argues here that sets and concepts exist external to any individual’s activities, and that questions about infinite sets are every bit as meaningful as questions about physics. This Platonistic strand in Gödel’s thought became more and more pronounced over the years, culminating in his 1964 addendum to the “What is Cantor’s Continuum Problem?” paper, which contains these passages:

Despite their remoteness from sense experience, we do have something like a perception of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don’t see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception … The set theoretical paradoxes are hardly any more troublesome for mathematics than deceptions of the senses are for physics…. Evidently the ‘given’ underlying mathematics is closely related to the abstract elements contained in our empirical ideas. It by no means follows, however, that the data of this second kind, because they cannot be associated with actions of certain things upon our sense organs, are something purely subjective, as Kant asserted. Rather, they, too, may represent an aspect of objective reality, but, as opposed to the sensations, their presence in us may be due to another kind of relationship between ourselves and reality.9

Despite his high level of scientific creativity during this period, Gödel seems to have had some opposition at the Institute, as he was not promoted from member to faculty status until 1953.10 It may have been that to some people Gödel’s Incompleteness Theorems seemed wholly negative, and that as a result they tried to dismiss these theorems as mere curiosities with no real mathematical or philosophical significance.

But in point of fact, Gödel’s Incompleteness Theorems are of an importance comparable to the Pythagoreans’ proof that the square root of two is irrational. The analogy is quite close. The Pythagoreans learned that no ratio of natural numbers could fully describe the relation between the diagonal and the side of a square. For his part, Gödel showed that no finitely describable theory can codify all mathematical truth. That is, he has shown that the set of all true statements about mathematics is finitely unnameable, and thus essentially random and infinite.

It is characteristic of Gödel’s work that the Incompleteness Theorems use purely mathematical reasoning to prove certain facts about the objective world. In 1949 he actually attempted to demonstrate the unreality of time by means of an argument in mathematical physics.11

Gödel published only one paper after this, a 1958 discussion of how one might prove the consistency of mathematics by assuming that mental objects have an objective existence.12 He did not like publicity and made at most two or three public appearances during the latter part of his life. Nevertheless, he continued as a guiding force in logic and set theory. Any mathematician who was invited to do so eagerly made the pilgrimage to his office at the Institute for Advanced Study.

In the next section, I describe my own pilgrimages.

### CONVERSATIONS WITH GÖDEL13

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I didn’t know where his real office door was, so I went around to knock on the outside door instead. This was a glass patio door, looking out on a little pond and the peaceful woods beyond the Institute for Advanced Study. It was a sunny March day, but the office was quite dark. I couldn’t see in. Did Kurt Gödel really want to see me?

Suddenly he was there, floating up before the long glass door like some fantastic deep-sea fish in a pressurized aquarium. He let me in, and I took a seat by his desk.

Kurt Gödel was unquestionably the greatest logician of the century. He may also have been one of our greatest philosophers. When he died in 1978, one of the speakers at his memorial service made a provocative comparison of Gödel with Einstein … and with Kafka.14

Like Einstein, Gödel was German-speaking and sought a haven from the events of the Second World War in Princeton. And like Einstein, Gödel developed a structure of exact thought that forces everyone, scientist and layman alike, to look at the world in a new way.

The Kafkaesque aspect of Gödel’s work and character is expressed in his famous Incompleteness Theorem of 1930. Although this theorem can be stated and proved in a rigorously mathematical way, what it seems to say is that *rational thought can never penetrate to the final, ultimate truth.* A bit more precisely, the Incompleteness Theorem shows that human beings can never formulate a correct and complete description of the set of natural numbers, {0, 1, 2, 3, …}. But if mathematicians cannot ever fully understand something as simple as number theory, then it is certainly too much to expect that science will ever expose any ultimate secret of the universe.

Scientists are thus left in a position somewhat like K. in *The Castle.*15 Endlessly we hurry up and down corridors, meeting people, knocking on doors, conducting our investigations. But the ultimate success will never be ours. Nowhere in the castle of science is there a final exit to absolute truth.

This seems terribly depressing. But, paradoxically, to understand Gödel’s proof is to find a sort of liberation. For many logic students, the final breakthrough to full understanding of the Incompleteness Theorem is practically a conversion experience. This is partly a by-product of the potent mystique Gödel’s name carries. But, more profoundly, to understand the essentially labyrinthine nature of *the castle* is, somehow, to be free of it.

Gödel certainly impressed me as a man who had freed himself from the mundane struggle. I visited him in his Institute office three times in 1972, and if there is one single thing I remember most, it is his laughter.

His voice had a high, singsong quality. He frequently raised his voice toward the ends of his sentences, giving his utterances a quality of questioning incredulity. Often he would let his voice trail off into an amused hum. And, above all, there were his bursts of complexly rhythmic laughter.

The conversation and laughter of Gödel were almost hypnotic. Listening to him, I would be filled with the feeling of perfect understanding. He, for his part, was able to follow any of my chains of reasoning to its end almost as soon as I had begun it. What with his strangely informative laughter and his practically instantaneous grasp of what I was saying, a conversation with Gödel felt very much like direct telepathetic communication.

The first time I visited Gödel it was at his invitation. I was at Rutgers University, writing my doctoral thesis in logic and set theory. I was particularly interested in Cantor’s Continuum Problem. One of Gödel’s unpublished manuscripts on this problem was making the rounds, and I was able to get hold of a Xerox of a Xerox of a Xerox.16

I deciphered the faint squiggles and thought about the ideas for several months, finally giving a talk on the manuscript at Rutgers. I had a number of questions about the proof Gödel had sketched and wrote him a letter about them.

He probably would not have answered—Gödel almost never answered letters. But I happened to be attending a weekly seminar at the Institute with Gaisi Takeuti, an eminent logician who was there for a year’s research. Gödel knew this, and one day while I was at the seminar in Takeuti’s office, he phoned up and asked that I come see him.

Gödel’s office was dim and unlit. There was comfortable carpeting and furniture. On the empty desk sat an empty glass of milk. Gödel was quite short, but his presence was such that visitors sometimes left with the impression that he was very tall.

When I saw him he was dressed as in all his pictures, with a suit over a warm vest and necktie. He is known to have worried a great deal about his health and was always careful to keep himself well bundled-up. Indeed, in the winter, one would sometimes see him leaving the Institute with a scarf wrapped around his head.

He encouraged me to ask questions, and, feeling like Aladdin in the treasure cave, I asked him as many as I could think of. His mind was unbelievably fast and experienced. It seemed that, over the years, he had already thought every possible philosophical problem through to the very end.

Despite his vast knowledge, he still could discuss ideas with the zest and openness of a young man. If I happened to say something particularly stupid or naive, his response was not mockery, but rather an amused astonishment that anyone could think such a thing. It was as if during his years of isolated thought he had forgotten that the rest of the human race was not advancing along with him.

The question of why Gödel chose to live most of his life in splendid isolation is a difficult one. Although he was not Jewish, the Second World War forced him to flee Europe, and this may have soured him somewhat on humanity. Yet, he loved life in America, the comfortable position at the Institute, the chance to meet Einstein, the great social freedom. But he spent his later years in an ever-deepening silence.

The first time I saw Gödel, he invited me; the second two times, I invited myself. This was not easy. I wrote him several times, insisting that we should meet again to talk. Finally I phoned him to say this again.

“Talk about what?” Gödel said, warily. When I finally got to his office for my second visit, he looked up at me with an expression of real dislike. But annoyance gave way to interest, and, after I’d asked a few questions, the conversation turned as friendly and spirited as the first. Still, toward the end of a conversation, when he was tired, Gödel would sometimes look at a visitor with an eerie mixture of fear and suspicion, as if to say, *what is this stranger doing in my retreat*?

Gödel was, first and foremost, a great thinker. The essence of the man is not to be found in his physical description, but rather in his ideas. I would like to describe now some of our discussions on mathematics, physics, and philosophy.

One of Gödel’s less well-known papers is a 1949 article called, “A Remark on the Relationship Between Relativity Theory and Idealistic Philosophy.”17 In this paper, probably influenced by his conversations with Einstein as well as by his interest in Kant, Gödel attempts to show that the passage of time is an illusion. The past, present and future of the universe are just different regions of a single vast space-time. Time is *part* of space-time, but space-time is a higher reality existing *outside of time.*

In order to destroy the time-bound notion of the universe as a series of evanescent frames on some cosmic movie screen, Gödel actually constructed a mathematical description of a possible universe in which one can travel back through time. His motivation was that if one can conceive of time-travelling to last year, then one is pretty well forced to admit the existence of something besides the immediate present.

I was disturbed by the traditional paradoxes inherent in time-travel. What if I were to travel back in time and kill my past self? If my past self died, then there would be no I to travel back in time, so I *wouldn’t* kill my past self after all. So then the time-trip would take place, and I *would* kill my past self. And so on. I was also disturbed by the fact that if the future is already there, then there is some sense in which our free will is an illusion.18

Gödel seemed to believe that not only is the future already there, but worse, that it is, in principle, possible to predict completely the actions of some given person.

I objected that if there were a completely accurate theory predicting my actions, then I could prove the theory false—by learning the theory and then doing the opposite of what it predicted. According to my notes, Gödel’s response went as follows: “It should be possible to form a complete theory of human behavior, i.e., to predict from the hereditary and environmental givens what a person will do. However, if a mischievous person learns of this theory, he can act in a way so as to negate it. Hence I conclude that such a theory exists, but that no mischievous person will learn of it. In the same way, time-travel is possible, but no person will ever manage to kill his past self.” Gödel laughed his laugh then, and concluded, “The *a priori* is greatly neglected. Logic is very powerful.”

Apropos of the free will question, on another occasion he said:

“There is no contradiction between free will and knowing in advance precisely what one will do. If one knows oneself completely then this *is* the situation. One does not deliberately do the opposite of what one wants.”

As well as questions, I also brought in for Gödel’s enjoyment some offbeat theories of physics I had come up with recently. I was quite satisfied when, after hearing one of my half-baked theories, he shook his head and said, “This is a *very* strange idea. A *bizarre* idea.”19

There is one idea truly central to Gödel’s thought that we discussed at some length. This is the philosophy underlying Gödel’s *credo*, “I do objective mathematics.” By this, Gödel meant that mathematical entities exist independently of the activities of mathematicians, in much the same way that the stars would be there even if there were no astronomers to look at them. For Gödel, mathematics, even the mathematics of the infinite, was an essentially empirical science.

According to this standpoint, which mathematicians call *Platonism*, we do not *create* the mental objects we talk about. Instead, we *find* them, on some higher plane that the mind sees into, by a process not unlike sense perception.

The philosophy of mathematics antithetical to Platonism is *formalism*, allied to positivism. According to formalism, mathematics is really just an elaborate set of rules for manipulating symbols. By applying the rules to certain “axiomatic” strings of symbols, mathematicians go about “proving” certain other strings of symbols to be “theorems.”

The game of mathematics is, for some obscure reason, a *useful* game. Some strings of symbols seem to reflect certain patterns of the physical world. Not only is “2 + 2 = 4” a theorem, but two apples taken with two more apples make four apples.

It is when one begins talking about infinite numbers that the trouble really begins. Cantor’s Continuum Problem is undecidable on the basis of our present-day theories of mathematics. For the formalists this means that the continuum question has no definite answer. But for a Platonist like Gödel, this means only that we have not yet “looked” at the continuum hard enough to see what the answer is.

In one of our conversations I pressed Gödel to explain what he meant by the “other relation to reality” by which he said one could directly see mathematical objects. He made the point that the same possibilities of thought are open to everyone, so that we can take the world of possible forms as objective and absolute. Possibility is observer-independent, and therefore real, because it is not subject to our will.

There is a hidden analogy here. Everyone believes that the Empire State Building is real, because it is possible for almost anyone to go and see it for himself. By the same token, anyone who takes the trouble to learn some mathematics can “see” the set of natural numbers for himself. So, Gödel reasoned, it must be that the set of natural numbers has an independent existence, an existence as a certain abstract possibility of thought.

I asked him how best to perceive pure abstract possibility. He said three things. i) First one must close off the other senses, for instance, by lying down in a quiet place. It is not enough, however, to perform this negative action, one must actively seek with the mind. ii) It is a mistake to let everyday reality condition possibility, and only to imagine the combinings and permutations of physical objects—the mind is capable of directly perceiving infinite sets. iii) The ultimate goal of such thought, and of all philosophy, is the perception of the Absolute. Gödel rounded off these comments with a remark on Plato: “When Plautus could fully perceive the Good, his philosophy ended.”

Gödel shared with Einstein a certain mystical turn of thought. The word “mystic” is almost pejorative these days. But mysticism does not really have anything to do with incense or encounter groups or demoniac possession. There is a difference between mysticism and occultism.

A pure strand of classical mysticism runs from Plato to Plotinus and Eckhart to such great modern thinkers as Aldous Huxley and D. T. Suzuki. The central teaching of mysticism is this: *Reality is One.* The practice of mysticism consists in finding ways to experience this higher unity directly.

The One has variously been called the Good, God, the Cosmos, the Mind, the Void, or (perhaps most neutrally) the Absolute. No door in the labyrinthine castle of science opens directly onto the Absolute. But if one understands the maze well enough, it is possible to jump out of the system and experience the Absolute for oneself.

The last time I spoke with Kurt Gödel was on the telephone, in March 1977. I had been studying the problem of whether machines can think, and I had become interested in the distinction between a system’s *behavior* and the underlying mind or consciousness, if any.

What had struck me was that if a machine could mimic all of our behavior, both internal and external, then it would seem that there is nothing left to be added. Body and brain fall under the heading of *hardware.* Habits, knowledge, self-image and the like can all be classed as *software.* All that is necessary for the resulting system to be alive is that it actually exist.

In short, I had begun to think that consciousness is really nothing more than simple existence. By way of leading up to this, I asked Gödel if he believed there is a single Mind behind all the various appearances and activities of the world.

He replied that, yes, the Mind is the thing that is structured, but that the Mind exists independently of its individual properties.

I then asked if he believed that the Mind is everywhere, as opposed to being localized in the brains of people.

Gödel replied, “Of course. This is the basic mystic teaching.”

We talked a little set theory, and then I asked him my last question: “What causes the illusion of the passage of time?”

Gödel spoke not directly to this question, but to the question of what my question meant—that is, why anyone would even believe that there is a perceived passage of time at all.

He went on to relate the getting rid of belief in the passage of time to the struggle to experience the One Mind of mysticism. Finally he said this: “The illusion of the passage of time arises from the confusing of the *given* with the *real.* Passage of time arises because we think of occupying different realities. In fact, we occupy only different givens. There is only one reality.”

I wanted to visit Gödel again, but he told me that he was too ill. In the middle of January 1978, I dreamed I was at his bedside.

There was a chess board on the covers in front of him. Gödel reached his hand out and knocked the board over, tipping the men onto the floor. The chessboard expanded to an infinite mathematical plane. And then that, too, vanished. There was a brief play of symbols, and then emptiness—an emptiness flooded with even white light.

The next day I learned that Kurt Gödel was dead.

### TOWARDS ROBOT CONSCIOUSNESS20

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*The human mind is incapable of formulating (or mechanizing) all its mathematical intuitions, i.e., if it has succeeded in formulating some of them, this very fact yields new intuitive knowledge, e.g., the consistency of this formalism. This fact may be called the ‘incompletability’ of mathematics. On the other hand, on the basis of what has been proved so far, it remains possible that there may exist (and even be empirically discovereable) a theorem-proving machine which in fact* is *equivalent to mathematical intuition, but cannot be* proved *to be so, nor even be proved to yield only* correct *theorems of finitary number theory.*

*—Kurt Gödel*21

For many years there has been debate over the precise significance of Gödel’s Incompleteness Theorem for the field of Artificial Intelligence.22 Gödel, especially in his later years, was a reclusive, even secretive, man, and the quotation printed above is very nearly the sum total of his published words on this important question. The purpose of this section will be to tease out the meaning and implications of that quotation.

The section breaks into four subsections. The first, which can be skimmed, describes precisely what is meant by a theorem-proving machine. The second develops the argument for the undefinability of truth a bit further, and shows how Gödel reaches the conclusion that humans can never write down a complete description of how they think about mathematics. The third subsection explains how machines too complex to design can nevertheless *evolve*, and the final subsection advocates a mystical answer to the question, “What is consciousness?”

#### FORMAL SYSTEMS AND MACHINES

Quite generally, a formal system is a set of symbols together with rules for employing them. A formal system has four components: (1) alphabet, (2) spelling and grammar, (3) axioms, and (4) rules of inference.

The alphabet is simply a supply of symbols. If one wishes to be quite abstract, he can get by with “0” and “1” as his only symbols. But normally one allows the upper-case and lower-case English and Greek alphabets, the punctuation symbols, the blank space, the usual logical symbols, the numerals and other mathematical symbols, and so on.

The spelling rules specify which strings of symbols are to be regarded as noun phrases (terms) and which strings are to be regarded as verb phrases (relations). The grammar rules specify which sorts of verbs and nouns can be meaningfully combined to make simple sentences, and how one can go on to build up compound sentences, sentences with quantifiers, and multi-sentence statements. A statement formed in accordance with the spelling and grammar rules is called well-formed.

The formal system singles out a certain set of well-formed statements as axioms or fundamental assumptions. The rules of inference specify the precise ways in which axioms can be changed and combined to “prove” the theorems of the formal system.

To be more precise, the well-formed statement *A* is said to be proved by the formal system if and only if there is a finite proof sequence *M _{1}*, …,

*M*of well-formed statements such that each of the

_{n}*M*

_{i}is either an axiom or is obtained from some of the previous

*M*’

_{j}*s*by one of the rules of inference, and such that the last sentence

*M*is actually

_{n}*A*. Now, the theorems of the formal system are those well-formed statements

*A*for which there exists a proof sequence ending with

*A*.

The theorems of a formal system are already latently present in the system’s axioms and rules of inference. Normally the formal system itself can be finitely described, but it will be able to prove infinitely many theorems. So a formal system can be thought of as a very compact way of summarizing a large body of fact.

All the theorems of classical mathematics, for instance, can be proved from the axioms and rules of inference of the formal system obtained by combining the propositional calculus, the predicate calculus, and the axioms of ZF (Zermelo-Fraenkel set theory). This formal system can be completely described in a few printed pages. There is, thus, a sense in which everything we know about mathematics is coded up in a few pages of print.

No other entire branch of science has allowed itself to be fully codified as a formal system yet, but there are many successful piecemeal efforts in this direction. The rules for how many electron shells a given atom has, the laws of heredity, the theory of electromagnetism, special relativity … all these can be expressed as formal systems yielding certain bodies of fact.

Usually the alphabet of a formal system will include something that can be used as a negation symbol. One normally only considers formal systems that are consistent, where this means that for no statement *S* can we obtain both *S* and the negation of *S* as theorems. This is a natural requirement because a formal system is, in practice, intended to summarize a collection of facts that obtain in some one possible world or part of a world … and it could never happen that *S* and the negation of *S* are both true facts.

Normally, one also requires that the formal systems we consider be put in an unambiguous and finitely describable form. When we say that a formal system is finitely describable, we mean that there are three definite finite procedures, WELL, AXIOM, and RULE, determining the system as follows.

Given any string *S* of symbols drawn from the system’s alphabet, WELL can be applied to determine whether or not *S* is a well-formed statement and AXIOM can be applied to determine whether or not *S* is an axiom. The finite procedure RULE can be applied to any well-formed statement in combination with any finite set of well-formed statements to determine whether or not the former follows from the latter according to any single rule of inference.

Without getting too technical, the essential aspect of the algorithmic procedures, WELL, AXIOM, and RULE, must be that they are (a) utterly mechanical in application and (b) always give an unequivocal YES or NO answer after some finite amount of time.

A finitely describable formal system *T* is sometimes best thought of as a theorem-listing machine. Given the formal system *T* based on the triple of algorithms (WELL, AXIOM, RULE), we can construct a machine *M _{T}* that will print out all of

*T*’s theorems, one after another.

Before describing *M*_{T}, we should first note that given a fixed alphabet there is an utterly mechanical procedure for generating all possible “words” or strings of symbols drawn from that alphabet. If, for example, the alphabet were the lower-case English alphabet, one could first list, in dictionary order, all one-letter words, then, in dictionary order, all two-letter words, then, in dictionary order, all three-letter words, and so on and on.

Now, *M _{T}* will operate by looping through the following list of instructions to generate three ever-growing stores

*St, We*and

*Th*, consisting, respectively, of possible strings, well-formed statements, and theorems. At the same time,

*M*will print the theorems out.

_{T}1. Generate the next possible string and add it to store *St.*

2. Check the most recent string with the procedure WELL. If YES, then add the string to the store *We.*

3. Check the most recent well-formed sentence with procedure AXIOM. If YES, then add the sentence to the store *Th.*

4. Use the procedure RULE to check, one by one, each member of *We* for derivability from the set *Th.* Add each sentence for which the answer is YES to the store *Th*, and print each of these new theorems on the output tape.

5. Return to (1).

There is a general theorem of recursive function theory stating that not only can any formal system be viewed as a machine, the converse is true as well. That is, given any digital computer *M* with unlimited memory, we can find a formal system *T _{M}* such that the possible outputs of

*M*are exactly the possible theorems of

*T*This holds not only for machines that simply print out long lists of theorems, but also for machines that exhibit branching behavior patterns, and even for machines that intereact with their environment.

_{M}.A typical machine is non-deterministic in the weak sense that it does have branching futures. *M* begins in an initial state (compare axioms), and then passes through a series of transitions according to programmed rules (compare rules of inference). *M* can be nondeterministic in the sense that for certain states *S* there are a variety of allowable “next states” I want to make the point that even if one allows the choice of next state to be made randomly (say, by counting clicks on a Geiger counter), the machine’s range of possible outputs is still equivalent to the set of possible theorems of some fixed formal system.

This is the case because, in and of itself, a formal system does not actually generate a list of its theorems. A formal system is, strictly speaking, a gestalt or starting state from which one can move out along various possible proof sequences to reach various possible theorems. The essential form of a formal system is thus more like a tree than a line. At the base of the tree we have the axioms and growing out from them we have all one-step proofs, and so on. At each node or fork of this tree is a theorem—a possible output.

Suppose, however, that we have a machine *M* that interacts with its environment. *M* then can be thought of as embodying a behavior function of the form *M(h, i) = o. “h*” stands for the history of what has happened to *M* since it was turned on, “*i*” stands for the stimulus or input at the present time, and “*o*” stands for a response or output that *M* may give to stimulus *i*, having had history *h.* Note that, as mentioned above, it is permissible that for given *h* and *i* there be several possible *o*’s. But now if we assume that the *h*’s, *i*’s, and *o*’s can be specified by strings of symbols, then it is not hard to see that the behavior of machine *M* can be coded up in some formal system *T _{M}* that has a variety of strings of the form “

*M*(

*h, i*) =

*0*” as its theorems.

#### THE LIAR PARADOX AND THE NON-MECHANIZABILITY OF MATHEMATICS

Let us now return to the idea of a theorem-listing machine, a finitely described device that prints out an endless list of statements. Obviously, the best possible theorem-listing machine would manage to print out all and only the true statements expressible in, let us say, the English language plus the customary printers’ symbols. Such a Universal Truth Machine would be a fine thing indeed. Once we had devised its finite procedures, WELL, AXIOM, and RULE, we could set it to running and simply lean back and watch the printout. If you were curious about the truth of some well-formed statement *S*, you would only have to sit there and wait until either *S*, or the negation of *S*, was printed out.

Before going any further, we should be a bit more precise about what sorts of string *S* will count as (well-formed) statements for which one can expect an unequivocal true or false decision.

A first point is that we do not necessarily expect a statement to consist of a single grammatical unit. That is, a statement can consist of several sentences, a whole paragraph, a book, or even a large number of books. We desire only that the entire welter of symbols embody a single assertion that is in fact either true or false. We will, however, not allow infinitely long strings of symbols as statements, for a statement should be, at least in principle, unambiguously and finitely communicable.

A second point is that a string such as “*x* is smaller than *y*” is not, in itself, a statement. For without being told what *x* and *y* are, no one could ever say that this string expressed either a truth or a falsehood. So a statement should not include any undefined terms or relations.

But now we have a rather nasty problem. What about the phrase “is smaller than” in the example above? Must the meaning of that too be explained before we have a definite statement? But if every term and relation, every noun and verb, must be explained, then we seem to face a real mess. A full statement will take on the form of an initial assertion, followed by definitions of the words used in the first definitions, followed by definitions of the words used in the second definitions, and so on.

Well … why not? Such are the joys of abstract speculation! By way of preventing unnecessary regresses, we specify that within any expanded statement, no word’s definition need be repeated. Since there are only finitely many words and word-creation schemata in English, the process should normally terminate. There may, however, be some cases in which the full expansion of a statement becomes infinite, and in such a case we will just have to write the statement off as non-well-formed, not really a statement.

Someone might object that a fully expanded statement will consist of circular sequences of definitions and will thus be useless. To this there are two responses. First, any circular definition more complicated than “*x* is *x*” does convey some information. Euclid’s circular definitions-of space as the set of all points and of a point as a region of space that has no parts-tells us that a point, whatever it is, cannot be made up of a number of component points. Second, the fact that one is using strings of symbols enables one to exhibit certain concepts without really having to define them. Take, for instance, the concept of the length of a string of symbols. “*xxx*” is a longer string than “*a*” and by multiplying such examples (perhaps schematically) the meaning of string length can be made evident.23

So let us expand our statements to be self-explanatory. When we are done, a statement such as “the square root of two is irrational” will have blossomed into a hefty volume of mathematics, and “all men are mortal” will have expanded to include a definition of most of the words in the English language.

But the simple listing of dictionary definitions will not always be enough to make a statement definite enough to be susceptible to a true/false determination. In order to evaluate a statement such as “Jimmy Carter is good,” one needs to know which criterion of “goodness” is intended. Does the statement assert that Jimmy Carter teaches Bible School, that Jimmy Carter should have been reelected, or that the flesh of Jimmy Carter is sweet and tasty? In filling the bare assertion out to a full statement, one of the many possible options must be chosen.

Let us return now to the concept of a UTM (Universal Truth Machine.) What we want is a finitely programmed device that will print out all the true statements, and no false statements. It is easy to prove that there can be no such UTM. (We repeat the argument from the section “Gödel’s Incompleteness Theorem.”)

For, say that you have a candidate UTM based on the triple of finite procedures (WELL, AXIOM, RULE). Now, since WELL, AXIOM, and RULE are entirely finite and mechanical, one should be able to construct a well-formed statement *G* saying, “The UTM based on the triple (WELL, AXIOM, RULE) will never print out this statement”24

If your UTM ever prints out *G*, then *G* will be false—so the UTM will have printed out a false sentence. But this would be impossible, as the UTM is to print out only true sentences. Therefore the UTM will never print out *G.* And therefore, *G* is in fact true. But now we have a true sentence that will never be printed out by our candidate UTM—which is therefore not truly universal.

Truth, in short, is not finitely describable. This fact provided the resolution of the Liar paradox discussed in “What is Truth?” Consider the string of symbols *B*, “This string of symbols does not express a well-formed statement that is true.” Clearly *B* cannot in fact be expanded to make a well-formed statement that is true—because if such a statement if true, then it isn’t. The reason *B* can’t be made well-formed is that if we expand *B* to include a complete explication of all the words it uses, then *B* will have to include a complete explication of what it means for a finite statement to be “true.” But, as we have just seen, this word “true” does not admit of any finite explication! So *B* is one of those statements that, if expanded to be completely understandable, becomes infinitely long. Now, an infinitely long string of symbols is not allowable as a statement, so there is really no paradox here. If *B* were a finite statement, then *B* would be true if and only if *B* were not true … but *B* is infinite, and that is the end of it. *B* gives the feel of a paradox, as does the similar string, “This sentence is not true,” but this is simply an illusion on a par with a man suspending himself by a piano wire and saying he can fly. In the case of *B*, the “piano wire” is infinitely long, and *B* is not really a statement at all.

So the concept of truth for finite statements is itself infinite. One might now be tempted to ask if there might be some higher notion of truth that can arbitrate the truth of every statement, whether finite, infinite, or transfinite? Not in this world. Looked at another way, the message of the Liar paradox is that one can never define truth in such a way as to apply to statements involving the concept of truth being defined.

Let me illustrate this with an example. There is in Rome a church, Santa Maria in Cosmedin, outside of which stands a huge stone disc. The face of the disc is carved in bas-relief to depict the visage of a hairy, bearded man. The mouth, located at about waist-level, is shaped something like a letter slot. Legend has it that God himself has decreed that anyone who sticks a hand in the mouth slot and then utters a false statement will never be able to pull the hand back out. But I have been there, and I stuck my hand in the mouth and said, “I will not be able to pull my hand back out.” (May God forgive me!)

It is perhaps not really so surprising that there is no set of rules that will suffice to generate all the possible truths. There is something about the fact that the rules exist in the world they attempt to describe that forbids this. What perhaps is surprising is that the same phenomenon also holds true for the small and orderly world of number theory.

The natural numbers, along with the equality relation and the operations of plus and times, do not seem very complicated. Even if one cannot get a finite description of all true statements, one might hope to get at least a finite description of all the true facts about natural numbers. But, according to Gödel’s 1930 Incompleteness Theorem, this is also impossible.

Gödel’s two Incompleteness Theorems state that all formal systems of a certain kind are subject to two related limitations. His results apply to any formal system *T* that is (i) finitely describable, (ii) consistent, and (iii) strong enough to prove the basic facts about whole-number arithmetic.

Gödel’s *First Incompleteness Theorem* states that no such formal system *T* is capable of deciding every statement about natural numbers. That is, for any such *T* there will be a fairly simple sentence about natural numbers such that neither this sentence nor its negation ever appears as a theorem of *T*. We already have seen the idea of the proof, as follows.

Let *M _{T}* be the theorem-listing machine that prints out all the theorems of

*T*. Suppose

*G*(

*T*) is the sentence saying, “

*M*will never print this sentence.” Now, as we saw in the discussion of Universal Truth Machines,

_{T}*G(T*) must in fact be a sentence that is true, but that

*M*never prints out.

_{T}The hard part is to show that *G*(*T*) can actually be put in the form of a sentence about natural numbers. Gödel did this by a process called Gödel-numbering, something like the coding process described in “The Library of Babel.” When Gödel is done, the self-referential sentence *G*(*T*) saying, “*T* cannot prove *G(T*),” has been converted into a sentence of pure mathematics stating that a certain polynomial equation has no solutions in the whole numbers.

The upshot is that any finitely describable consistent theory *T* provides an incomplete description of the natural numbers; for given any such *T* there will be a sentence *G*(*T*) about natural numbers that *T* can neither prove nor disprove.25

Now for the *Second Incompleteness Theorem.* Recall that the condition, “*T* is consistent” means that for no sentence *S* does *T* prove both *S* and the negation of *S.* This condition can be abbreviated as Con(*T*), and by using Gödel-numbering again we can convert Con(*T*) into a purely number-theoretic sentence, once again a sentence saying that a certain Diophantine equation has no solution.

Insofar as the theory *T* embodies a correct description of the mathematical universe, we might expect Con(*T*) to be a fairly obvious and readily deducible fact. But the Second Incompleteness Theorem tells us that if *T* is a theory satisfying conditions (i) through (iii), then *T* cannot prove Con(*T*).

The proof works by going back and formalizing the proof of Gödel’s First Incompleteness Theorem inside the system *T.* Once we do this, we have proved “Con(*T*) implies *G(T)”.* But now, since *T* cannot prove *G*(*T*), it must be that *T* cannot prove Con(*T*) either!

We are now in a position to understand the first part of the quote from Gödel that stands at the head of this paper. As mentioned above in “Conversations with Gödel,” Gödel took the viewpoint that the natural numbers, the transfinite sets, and all other mathematical objects are immaterial, but actually existing, entities. By a process that Gödel called mathematical intuition, humans come to learn certain facts about the universe of mathematics, and we also come to learn certain correct methods of reasoning from given facts to further facts. Mathematical intuition is to be thought of as a process just as reliable as ordinary sense perception.26

Now, since the facts and methods of reasoning that we learn are correct descriptions of an actually existing mathematical universe, there is no possibility of ever producing a contradiction in mathematics. On the basis of true mathematical facts and correct mathematical reasoning, we will never find ourselves proving that, say, zero is not equal to zero. In other words, whenever we can produce a description *K* of our present body of mathematical knowledge, then we can be sure that Con(*K*) is true.

Now, say that we are able to come up with a finite description of a formal system *K* that summarizes our entire present body of mathematical knowledge. *On the one hand*, by the considerations of the last paragraph, we know that Con(*K*) is true. On the basis of our finite description of *K* we can even apply Gödel numbering and come up with a specific true number-theoretic sentence Con*(K).* But, *on the other hand*, since *K* satisfies conditions (i) through (iii), the Second Incompleteness Theorem tells us that *K* cannot prove Con(*K*)!

It is for this reason that Gödel says the human mind is incapable of mechanizing all of its mathematical intuitions. For to mechanize our intuitions is to produce a finite description of a formal system *K.* But as soon as we see this finite description, our mathematical intuition shows us a fact, Con(*K*), which the mechanized system does not prove. So it is not true that the mechanized system *K* proves all facts that we can perceive through our mathematical intuition.

#### ARTIFICIAL INTELLIGENCE VIA EVOLUTIONARY PROCESSES

Let us now consider the second part of Gödel’s remark. He remarks that even though we cannot write the program for a theorem-proving machine that is equivalent to human mathematical intuition, it is possible that such a machine could exist and even be empirically discoverable.

Let us suppose that there is a machine *R* that is equivalent to human mathematical intuition. A first fact to be established is that we could never understand *R’s* program. Let us make this point quite clear.

(i) Being a theorem-proving machine, *R* is finitely describable. (ii) Since *R* is equivalent to our *a priori* consistent mathematical intuitions about the mathematical universe, *R* is consistent, (iii) For the same reasons, *R* is of course strong enough to prove the basic facts of whole-number arithmetic. Therefore, by the Second Incompleteness Theorem, *R* cannot prove Con*(R).*

Now, since *R* is in fact exactly equivalent to human mathematical intuition, it must be that the humans can never prove Con(*R*) either. This can only be because the humans can never understand the finite description of *R* well enough to prove that *R* is, in fact, equivalent to their own mathematical intuitions, which are known, by a sort of higher-level intuition, to be consistent. The grounds for our lack of understanding of *R’s* program will, presumably, lie in the program’s length and subtlety.

But how could such an utterly incomprehensible machine actually be produced, when we know that we could never understand it, let alone build it?

The answer is evolution. Before his premature death in 1957, John von Neumann worked out the basic theory of self-reproducing automata.27 There is, in principle, no difficulty in designing robots that are capable of building factories to produce other robots. There is also no difficulty in arranging things so that the old robots can copy their own programs onto the processors of the new robots. It is simply a matter of assembling the hardware, and then replicating one’s software onto the new hardware.

A current IBM design goal is the construction of super-cooled computers that will fit inside an 8 × 8 × 10 centimeter box, and have a 256,000 word cache memory (workspace) and a 64 million word main memory. Each bit will be represented by a quantum of magnetic flux generated by a persistent current in a super-conducting Josephson junction. It is perhaps not too much to hope that the following generation of computers will be able to enter the 1010 word range characteristic of human brains.

Now imagine equipping a few thousand robots with these liquid-helium-temperature silicon brains and setting them loose on the moon. Their prime directive is to mine, smelt, fabricate, and assemble the materials necessary to build more robots. They find the moon congenial: the low temperatures, the abundant solar energy, the lack of corrosive water vapor or gaseous oxygen, the abundance of silicon. They are fruitful, and they multiply.

Assuming that they are programmed to place high priority on *self-reproduction*, there will be an inevitable competition for the raw materials and finished supplies with an attendant effect of *natural selection.* In addition, we can ensure that their programs undergo regular *mutations.* This could be done by placing somewhere in the base program an imperative never to copy oneself exactly. Each time a program was transferred, a substantial number of changes would be made in it, these changes to be determined randomly, let us say by counting cosmic rays, or simply by waving a powerful magnet over the new scion’s head.

Actually, a large part of evolutionary diversity arises not from actual mutating gene change, but rather from the shuffling of genes inherent in sexual reproduction. Presumably something along these lines could be arranged, with two (or more) robots “sexually reproducing” together by pooling their hardware resources and shuffling together various of their subprograms to produce a new program for the little scion.

Rather than immediately sending billions of dollars worth of hardware to the Moon, one would in practice begin a project of this nature in the laboratory. It would not even be initially necessary to deal with mobile hardwares capable of physical self-reproduction. Instead, as a first stage, I would envisage some thousands of AI programs competing (perhaps on the basis of scores on certain tests) for the right to be replicated, with one copy always incorporating some randomly determined mutations. It seems possible that the process could be speeded up enough for significant evolutionary effects to emerge after only a few years run-time.

Once the programs had gotten intricate enough to be incomprehensible, or understandable only with great difficulty, they could be packed into the factory-building robots and shipped off to the Moon. The economic incentive for this is, of course, that we would like to be able to exploit the Moon’s resources, and it is possible that robots would have a better cost to productivity ratio than human colonists.28

As the robots on the Moon continue to evolve, there is always the danger that an unfortunate series of lethal mutations might force them into extinction. One would like perhaps to guard against lethal mutation by leaving certain core life-support sections of their programs inviolable. A problem with this is that a mutation that is lethal now may not be lethal to future variants of the species, and vice versa. Lungs, for instance, are very bad for a fish, but very good for an amphibian.

But assuming all goes well, that God, perhaps, will guide the robots’ evolution as He may have guided ours, then we could expect that there would eventually be a large and autonomous robot civilization on the Moon, and that some of the robots there might interest themselves in mathematics. It is at this point that there is a real possibility that there could arise a theorem-proving machine *R* (a robot mathematician) whose abilities are, in fact, equal to the resources of human mathematical intuition.

There is no reason why *R* could not print out his program for us, perhaps in some extremely compact and coded form (compare sperm cells!). But, as was discussed above, we would not be able to understand this program, nor would we be able to prove to our satisfaction that *R* was consistent. The interesting thing is that even though we cannot understand the program of *R*, we are able to set up the physical conditions that lead to *R*’s coming into existence.

#### ROBOT CONSCIOUSNESS

With the paradigm of the last subsection in mind, it now seems evident that there could be robots whose general behavior was the same as the behavior of human beings. These robots would be thinking beings who had evolved on a substrate of metal and silicon chips, just as we are thinking beings who have evolved on a substrate of amino acids and other carbon-based compounds. Would one be justified in saying that these highly evolved robots possess consciousness in the same sense that humans do?

Upon lengthy introspection, most people will agree that the individual person consists of three distinct parts: (a) the hardware, the physical body and brain; (b) the software, the memories, skills, opinions, and behavior in general; (c) consciousness, the sense of self or personal identity, pure awareness, the spark of life, or even the soul.

I would like to argue that any component of parts (a) or (b) can be replaced or altered without really affecting (c). My purpose in arguing this way will be to show that there is nothing about part (c) that is specific to the individual.

Let us begin with the hardware. If one gets an artificial leg, kidney, or heart, one is still the same person. I maintain that it is possible to envisage a time when one could even get a new artificial brain. This could be done by, let us say, holographically recording the physical, electrical, and biochemical structure of the brain, and then transferring this structure isomorphically onto a large silicon-chip system or onto some appropriate module of culture-grown tissue. Presumably one would experience such a transfer only as a brief period of unconsciousness after which one would go on thinking much the same as one had before. The whole process would be comparable to putting a given program into a new computer.29

Now for the software. It is an utterly familiar feeling to look back on the way one was behaving a year, a month, or an hour ago and to be amazed. One’s personality is always changing, and one is always learning new things and forgetting old ones. There is also the extreme example of brainwashing. We are inclined to say that a person’s essential identity is unchanged, even if he has been given a completely false set of memories.

What then remains for part (c)? I contend that the sum total of the individual consciousness is the bare feeling of existence, expressed by the primal utterance, *I am*. Anything else is either hardware or software, and can be changed or dispensed with. Only the single thought *I am* ties me to the person I was twenty years ago.

The curious thing is that you must express your individual consciousness in the same words that I use: *I am. I am me. I exist.* The philosopher Hegel was very struck by this fact, and deemed it an instance of “the divine nature of language.”

What conclusion might one draw from the fact that your essential consciousness and my essential consciousness are expressed in the same words? Perhaps it is reasonable to suppose that there really is only one consciousness, that individual humans are simply disparate faces of what the classical mystic tradition calls the One.

But we can go farther than this. The essence of consciousness is, really, nothing more than simple existence. *I am.* Why should the possession of this sort of consciousness be denied to anything that does exist? Aquinas has said that God is pure existence unmodified. Is it not evident that there is a certain single something–call it God, or the One, or pure existence–that pervades the world as it is? Consider the Zen phrasing of this: *The universal rain moistens all creatures*.30 Or think of the world as a stained-glass window with light shining through every part.

To exist is to have consciousness. The other things one might feel are necessary for consciousness are more or less complicated sorts of hardware and software, patterns of mass and energy. But no pattern can be conscious until it exists, until it is brought into reality. Existence is, finally, the only thing required for consciousness. A rock is conscious. This piece of paper is conscious. And so, of course, is a robot, both before and after his behavior evolves to our level.

Traditionally, those who have asserted the equivalence of men and (possible) machines have been positivists, mechanists, materialists. They put their viewpoint this way: “Men are no better than machines.” But if one only changes the emphasis, then this equivalence can become the expression of a deep belief in the universality and reality of consciousness: “Machines can be as good as men!”

### BEYOND MECHANISM?

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There seem to be three possible viewpoints regarding questions about human and robot souls. *Mechanism:* Neither men nor robots are anything but machines, and there is no reason why man-like machines cannot exist. *Humanism:* Men have souls and machines do not, therefore no robot can be quite like a man. *Mysticism:* Everything, whether man or machine, participates in the Absolute, therefore it should be possible for man-like machines to exist. Perhaps “mysticism” is not really a good name for this last view, but let it stand.

In the last section I argued for the third viewpoint, under which “having a soul” is a concept automatically satisfied by anything that exists. But, since this view leads to a conclusion identical to that of mechanism, the reader may feel that I have ducked the real issue. One feels oneself to be “more than just a machine.” Is there any possible justification for this belief short of recourse to an all-pervading Absolute?

In two classic papers, Alan M. Turing developed a powerful argument for mechanism: All that we can know about another person’s mind is based on observing his behavior (i.e., by conversing with him, reading his writings, and so on); and there seems to be, in principle, no reason why there could not be a machine whose “conversation” is just like a person’s.31

Now, it might be objected that such an argument does not account for private mental phenomena, such as mental images, purposes, emotions, and the like. But a determined mechanist can say that what we call a mental image is merely a model or simulation such as computers often use, that a purpose is simply an assignment of utility values to certain internal states, and that emotions are just ways of assigning values to external phenomena as well.

An even stronger argument for the mechanist position can be formulated by asserting that: 1) The activities of the mind are isomorphic to certain electro-chemical processes in the brain; and 2) The brain functions basically like a digital computer.32

These assertions can be restated: 1) There is no mind apart from matter; and 2) The brain is finite. The idea is that, given 1) and 2), we can be sure that the operation of the mind is a finite law-like process; and any such process can, in fact, be modelled by a large enough digital computer.

Assumption 1 is often questioned by those who believe in telepathy and other forms of ESP. These investigators have pointed out various unusual occurrences or perceptions that do not seem to fit in with the idea that the mind is simply a phenomenon taking place in the limited confines of the skull. I really do not know what to make of the claims made for ESP. Like any other person, I have experienced my share of what Jung calls *synchronicities:* meaningful concidences, true premonitions, and lucky guesses.33 But the desire for results is so strong, and the opportunities for deception so great, that the most extreme caution is necessary.

One of the things most lacking in ESP research is any reasonable theory of how the mind could, in fact, go beyond the confines of the brain.34 As I suggested at the end of Chapter 2, it would be nice if Cantor’s two-substance theory were true, for then we could perhaps have the mind or “astral body” made up of some higher-level substance quite different from matter. But there is no evidence for such a theory—it is still only an idea for an idea.

A recent development in physics that could lead to some interesting developments is the experimental disproof of the Bell Inequality.35 What the experiments indicate is that there is a very real sense in which particles that have interacted continue to affect each other long after the interaction has taken place … and in an instantaneous way! If this is indeed true, then the universe would of necessity behave like a single organic whole, leading to the possibility of choosing to identify one’s mind with the cosmos rather than with some individual brain. But, as I pointed out at the end of the last section, there is no reason why such a form of higher consciousness would not be open to robots as well. Of course, then robots would not be “machines” in the narrow, finitistic sense intended by the mechanists.

Going back to the argument for mechanism, what about assumption 2? As was discussed in the section “Infinities in the Small,” it is certainly possible that matter is infinitely divisible. If this were indeed true, then any material object, such as a human brain, would in fact be infinitely complex. Perhaps we really do think infinite thoughts, and it is simply an accidental property of this limited scale level that our descriptions of them come out finite. Sometimes I actually am able to believe this for a few minutes, keeping in mind Cantor’s remark that “the infinite even inhabits our minds.”36

“I can’t believe *that*!” said Alice.

“Can’t you?” the Queen said in a pitying tone. “Try again: draw a long breath, and shut your eyes.”

Alice laughed. “There’s no use trying.” she said: “one *can’t* believe impossible things.”

“I daresay you haven’t had much practice,” said the Queen. “When I was your age, I always did it for half-an-hour a day. Why sometimes I’ve believed as many as six impossible things before breakfast.”37

Perhaps the mechanists are right. But it is certainly worthwhile to keep an open mind, even if this means occasionally seeing how it feels to believe something that is “impossible.”

### PUZZLES AND PARADOXES

*(Answers, p.*(*301*)

1. Consider the sentence *S*: “THIS SENTENCE CAN NEVER BE PROVED.” Show that if *S* is meaningful, then *S* is not provable, and that therefore you can see that *S* must be true. But this constitutes a “proof” of *S.* How can the paradox be resolved?38

2. Consider the following argument for the unreality of death: a) A person’s mind and personality is equivalent to his software, that is, to the programming of his brain; b) Any software structure can be coded up by some large set of natural numbers; c) Every set of numbers exists eternally as a mathematical abstraction independent of the physical universe; d) Therefore each individual’s personality is immortal. Why is it that this type of immortality does not seem like enough?

3. Consider this somewhat stronger argument for the immortality of artists: a) In a great work of art, an artist codes up a large part of his software, his personal feelings about life; b) When one deeply immerses oneself in a work of art, one takes on, for a few moments, the actual software coded up by that work; c) Therefore every time one truly appreciates a work of art, one is, for the moment, isomorphic to the artist, and the artist is thus (briefly) reincarnated over and over. If there were, a hundred years from now, someone exactly like you, would it make a difference to you? Would this be any different from being frozen for a hundred years and then resuscitated?

4. Consider the following fanciful example of self-reproduction (due to Doug Hofstadter). “Imagine that there is a nickelodeon in the local bar that, if you press buttons 11-U, will play a song whose lyrics go this way: *Put another nickel in, in the nickelodeon,/All I want is 11-U, and music, music, music*.” Assuming that everything goes well, some state of some system will occur over and over here. Specify the state and the complete system.

5. There is a word game sometimes called “word golf” that embodies a simple analogy of a formal system. One starts with a given word (say, LOVE) that serves as the initial “axiom.” The “rule of inference” is that at each step one can change one letter of the word in hand, provided the change produces a new English word. The “theorems” following from a given “axiom” word are those words that can be reached by a series of transformations according to the rule of inference. Thus HATE is a theorem, or consequence, of LOVE. The “proof” of this is the sequence LOVE, ROVE, RAVE, HAVE, HATE. The reason it is called “word golf” is that one tries to find “proofs” with the shortest possible number of steps. If you allow the somewhat obscure word LAVE, then the derivation above can be shortened by a step to LOVE, LAVE, LATE, HATE. How many steps do you need to turn COLD into WARM, BEER into WINE, FISH into FOWL?

6. In this chapter we discussed a UTM (Universal Truth Machine) that would print out all true statements. In the section “What is Truth,” a sort of TSM (Truth Sorting Machine) was discussed. The TSM was to look at any book and decide, after a longer or shorter interval of time, whether or not the book was true. We have proved that neither a UTM nor a TSM can exist. *BUT*, ignoring this fact for now, show that, with a few modifications, any UTM could be converted into a TSM, and any given TSM could be converted into a UTM.

7. Present day computers are, both with respect to hardware and with respect to software, far superior to the computers of thirty years ago. Show that this improvement can be viewed as a sort of robot evolution, and point out what processes have played the roles of reproduction, selection, and mutation.

8. The passage from Gödel quoted at the beginning of “Towards Robot Consciousness” goes on to make a further point: “Either the human mind surpasses all machines (to be more precise, it can decide more number-theoretical questions than any machine) or else there exist number-theoretical questions undecidable for the human mind.”40 Explain how this is really just a restatement of the result that no machine can answer all number-theoretical questions.

## CHAPTER FIVE

## THE ONE AND THE MANY

Think of Everything–the whole physical universe past and present, all the other possible universes, all the possible thoughts, all the mathematical sets–or *can* one think about it? The classical One-Many problem is this question: Can Everything be regarded as a unity, as a single definite thing? Is the world a One or a Many?

The first section of this chapter contains a brief account of some aspects of the classical One/Many problem. The problem allows itself to be stated in terms of set theory, so the following section, “What Is a Set?,” introduces the general notion of set, and “The Universe of Set Theory” describes various ways in which the One/Many problem arises in set theory. The final section of the chapter, “Interface Enlightenment,” describes a certain sort of metasolution to the problem, where *meta* solution means a solution obtained by looking at the problem from a higher level standpoint. Such solutions are not always satisfying—one type of metasolution to a chess problem is to overturn the board!

### THE CLASSICAL ONE/MANY PROBLEM1

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There are two forms of the One/Many problem: i) How many *kinds* of things are there; ii) How many *things* are there? The natural first answer is that there are many different kinds of things and many different things.

There is, however, a perennial desire to reduce the world’s diverse phenomena to a single basic kind, to believe that ultimately all things are built of the same stuff. Matter, sensation, thought, and form have all been candidates for *Urstoff.* The belief that there is ultimately only one kind of thing in the world is called *monism of kinds.* Materialism and idealism are both monisms of kinds; the monism of kinds that asserts that everything is a set will be considered in “The Universe of Set Theory.”2

Instead of uniting things from the bottom up, one can work from the top down, starting with the assertion that “All is One.” *Monism of substance* asserts that everything is a part or manifestation of a higher unity that is usually called the Absolute.

It is, of course, obvious that the word or concept “Everything” serves to form the world at least superficially into a One. In the same way, the bare concept “set” makes a One of the universe of set theory, but without answering the real question of whether this universe is in any sense a definite completed object. The heart of question ii) is whether or not the world is One in some organic sense, rather than in the sense of mere wordplay.

Perhaps the principal reason for believing that the world *is* an organic One is the sort of mystical insight that Lovejoy somewhat slightingly refers to as “monistic or pantheistic pathos.”3 The fact that it is occasionally possible to feel an all-encompassing unity in the world is, however, not conclusive, as it is equally possible to feel a diversity in the world that defies unification.

It is possible to *argue* for monism of substance in various ways. One idea is that ultimately everything in the world is related to everything else, and that the Absolute is the means or essence of this interrelatedness. Here the Absolute serves as a sort of connective tissue that fixes the individuals of the world into their perceived relational structure.

Another approach is to argue that any two things are, in a sense, the same; and that the Absolute is the one endlessly diversifying thing that exists.

But it is at least questionable whether such an Absolute actually exists, and pluralism of substance remains a reasonable position. This position is forcefully presented by William James in *A Pluralistic Universe:*

“… the pluralistic view which I prefer to adopt is willing to believe that there may ultimately never be an all-form at all, that the substance of reality may never get totally collected, that some of it may remain outside of the largest combination of it ever made, and that a distributive form of reality, the *each-*form, is logically as acceptable and empirically as probable as the all-form commonly acquiesced in as so obviously the self-evident thing.”4

Not to keep the reader in suspense, I should come out with my position on the One/Many problem. But this is not so simple. In certain ways I agree with James, that there really is no ultimate single universe. But on the other hand, I do feel that the simple predicate, “exists,” does tie everything together into a unity that it is, in principle, possible to experience directly. Rationally the universe is a Many, but mystically it is a One. The question that really interests me is this: How do we reconcile the Absolute as One with the Absolute as Many? How do we fit together the world of feeling and the world of thought?

But the reader should not hope for any final, tidy answer to this aspect of the One/Many problem. No one knew more about it than Plato. Most of the *Parmenides* and the *Sophist* deal with the One and the Many. In section 15 of *Philebus*, Plato’s last dialogue, he has the aging Socrates deliver this wry, weary, wise summary of what he knows about the One and the Many:

We say that the one and many become identified by thought, and that now, as in time past, they run about together, in and out of every word which is uttered, and that this union of them will never cease, and is not now beginning, but is, as I believe, an everlasting quality of thought itself, which never grows old. Any young man, when he first tastes these subtleties, is delighted, and fancies that he has found a treasure of wisdom; in the first enthusiasm of his joy he leaves no stone, or rather no thought, unturned, now rolling up the many into the One, and kneading them together, now unfolding and dividing them; he puzzles himself first and above all, and then he proceeds to puzzle his neighbours, whether they are older or younger, or of his own age that makes no difference; neither father nor mother does he spare; no human being who has ears is safe from him, hardly even his dog, and a barbarian would have no chance of escaping him, if an interpreter could only be found.5

### WHAT IS A SET?

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It is impossible to improve upon Cantor’s succinct 1883 definition: “A set is a Many which allows itself to be thought of as a One.”6 One of the most basic human faculties is the perception of sets. Consider this random pattern of X’s.

As you idly gaze at the picture, your brain searches out and notices various subpatterns, various constellations of X’s. On the upper right is a rough hexagon. Running from lower left to upper right is an approximate line. Running from upper left to lower right is a double camel’s hump of sorts. Triangles, a question mark, a tennis racquet, and so on.

When you think about your associates and acquaintances you tend to organize your thought by sorting these people into overlapping sets: friends, women, scientists, drinkers, gardeners, sports fans, parents, etc. Or the books that you own, the recipies that you know, the clothes you have—all of these bewildering data sets are organized, at the most primitive level, by the simple and automatic process of set formation, of picking out certain multiplicities that allow themselves to be thought of as unities.

Do sets exist even if no one thinks of them? The numbers 2, 47, 48, 333, 400, and 1981 have no obvious property in common, yet it is clearly *possible* to think of these numbers all at once, as I am now doing. But must someone actually think of a set for it to exist?

This is really a version of the old chestnut, *does a falling tree make a noise if there is no one there to hear it*? The “tennis racquet” was present in the pattern of X’s, even before I noticed it. By circling it, I do not bring it into existence, I merely point it out as an objectively existing feature of the external world.

Even if no one ever notices some given set, it still exists as a certain possible thought or perception. In the same way, the lonely falling tree’s sound exists as a certain possible perception, a condition of the world expressible as, *if a person were present he would hear a crash.*

The basic simplifying assumption of Cantorian set theory is that sets are there already, regardless of whether anyone does or could notice them. A set is not so much a “Many which allows itself to be thought of as a One” as it is a “Many which would allow itself to be thought of as a One, if someone with a large enough mind cared to try.” For the set theorist, the bust of Venus is already present in every block of marble. And a set *M* consisting of ten billion random natural numbers exists even though no human can ever see *M* all at once. *A set is the form of a possible thought*, where “possible” is to be taken in the broadest sense.

At this point one might well ask if there is anything that is not a set, or if there are any “thoughts” that are not possible. The answer, surprisingly, is yes. We can imagine that some sets might be members of themselves, and some might not be. So one might expect to be able to discuss a set *R* defined to be the *set of all sets that are not members of themselves.* In symbols we would have *R* = {*x:x* ∉ *x*}.

On the surface there seems to be no reason why *R* should not be a Many which allows itself to be thought of as a One. But now ask this question: Is *R* a member of *R*? Is *R* ∈ *R*?

A little thought makes it evident that if *R* is a member of *R*, then *R* (not being a set that is not a member of itself) is not a member of *R.* And if *R* is a set and not a member of *R*, then *R* (being a set that is not a member of itself) is a member of *R*.

So *if R* is a set, then we have a contradictory state of affairs with *R* being in itself if and only if it is not in itself. So we are forced to conclude that *R* is, in fact, *not a set. R* is a Many which does not allow itself to be thought of as a One.

But why not? Why can’t *R* be a set? The reason seems to be that *R* is too big.

In set theory we normally assume that no set is a member of itself anyway. This principle, called the *genetic formation of sets principle*, embodies the idea that a set *S* exists by virtue of two logical steps: i) a bunch of possible members *x* are in existence; ii) some of these *x’s* can be combined into a unity to form *S.* The idea is that if we had *S* ∈ *S*, then trying to separate out the steps i) and ii) would lead to an infinite regress, with i) depending on ii) (since *S* is one of the possible members *x* of *S*), and with ii) depending on i) (since *S* can’t exist unless its members exist). There is nothing directly contradictory here, but it is certainly more pleasant to speak only of sets that are logically built up from simpler sets and objects.

Note that if no set is a member of itself, then the collection *V* of all sets is the same as the collection *R* of all sets that are not members of themselves. Now we know that *R* is not a set, so if *R = V*, then we also know that *V* is not a set. This is one way to prove that the universe of set theory is a Many that does not allow itself to be thought of as a One.

Of course, since we have stipulated that no set is a member of itself, there is a simpler way to see that *V* is not a set. For if *V* were a set we would have *V* being a set that is a member of itself (since every set is a member of *V*). So the reason that *R* is not a set is not because its definition has something to do with self-reference, but rather, that *R* is big like *V*, the *class of all sets.*

Set theorists use the expression “class” to mean a collection or multiplicity of any kind. A class may or may not be unifiable into a set. If not, we call it a *proper class.* Thus, *V* is a proper class. *V* is a Many which does not allow itself to be thought of as a One.

Cantor was well aware of the distinction between sets and collections that are proper classes. He wrote about it in a famous 1899 letter to Dedekind:

If we start from the notion of a definite multiplicity of things, it is necessary, as I discovered, to distinguish two kinds of multiplicities.

For a multiplicity can be such that the assumption that *all* its elements ‘are together’ leads to a contradiction, so that it is impossible to conceive of the multiplicity as a unity, as ‘one finished thing.’ Such multiplicities I call *Absolutely Infinite* or *inconsistent multiplicities.*

As we can readily see, the ‘totality of everything thinkable,’ for example, is such a multiplicity …7

The reader may recall that Cantor’s last point was discussed in the section “The Absolute Infinite.” If we assume that “thought” means “rationally communicable thought built up from simpler forms,” then it is evident that for any universe *T* of possible thoughts, the thought of *T* as a whole will be a new thought not lying in *T*. So any attempt to think of everything thinkable leads to an (Absolutely) infinite sequence of approximations that do not seem to converge to anything defi