At one point, we turned to John Conway and the surreal numbers, and so please enjoy this excerpt from the conversation.

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Lex Fridman(02:46:47) So speaking of the land of nonsense, I have to ask you about surreal numbers, …there’s this aforementioned wonderful blog post on the surreal numbers and that there’s quite a simple surreal number generation process that can basically construct all numbers. So maybe this is a good spot to ask what are surreal numbers and what is the way we can generate all numbers?

Joel David Hamkins(02:47:20) So the surreal number system is an amazing, an amazingly beautiful mathematical system that was introduced by John Conway.

Lex Fridman(02:47:30) Rest in peace, one of the great mathematicians ever on this earth.

Joel David Hamkins(02:47:33) Yes, absolutely. And I really admire his style of mathematical thinking and working in mathematics and the surreal number system is a good instance of this. So the way I think about the surreal numbers system is what it’s doing is providing us a number system that unifies all the other number systems. So it extends the real numbers. Well, not only does it extend the integers, the natural numbers, the rational numbers, and the real numbers, but also the ordinals and the infinitesimals. So they’re all sitting there inside the surreal numbers, and it’s this colossal system of numbers. It’s not a set even. It’s a proper class, it turns out, because it contains all the ordinal numbers.

Joel David Hamkins(02:48:19) But it’s generated from nothing by a single rule, and the rule is, so we’re going to generate the numbers in stages, in a transfinite sequence of stages. And at every stage, we take the numbers that we have so far and in all possible ways, we divide them into two sets, a lower set and an upper set, or a left set and a right set. So we divide them into these two sets so that everything in the left set is less than everything in the right set, and then at that moment, we create a new number that fits in the gap between L and R. Okay? That’s it. That’s all we do. So let me say it again.

Joel David Hamkins(02:49:05) The rule is we proceed in stages, and at any stage, in all possible ways, we divide the numbers we have into two collections, the left set and the right set, so that everything in the left set is less than everything in the right set. And we create a new number, a new surreal number that will fit in that gap. Okay. So for example, we could start… Well, at the beginning, we don’t have any numbers. We haven’t created anything yet, and so, we could take nothing and we could divide it into two sets, the empty lower set and the empty upper set. I mean, the two empty sets. And everything in the empty set is less than everything in the empty set because that’s a vacuous statement.

Joel David Hamkins(02:49:48) So we’re, we satisfy the conditions and we apply the number generation rule, which says we should create a new number. And this is what I call the big bang of numbers, the surreal genesis when the number zero is born. Zero is the firstborn number that is bigger than everything in the empty set and less than everything in the empty set. Okay, but now we have this number zero, and so therefore, we now can define new gaps. Because if we put zero into the left set and have an empty right set, then we should create a new number that’s bigger than zero and less than everything in the empty set, and that number is called the number one.

Joel David Hamkins(02:50:30) And similarly, at that same stage, we could have put zero into the right set, and so that would be the firstborn number that’s less than zero, which is called minus one. So now we have three numbers, minus one, zero, and one, and they have four gaps because there could be a number below minus one or between minus one and zero or between zero and one or above one, and so we create those four new numbers. The first number above one is called two. The first number between zero and one is called 1/2, and then on the negative side, we have minus 1/2 and minus two and so on. So now we have, what is that, seven numbers. So there’s eight gaps between them.

Joel David Hamkins(02:51:10) So at the next birthday, they call them, the next stage will be born all the numbers between those gaps, and then between those and between those and so on. And as the days progress, we get more and more numbers. But those are just the finite birthdays, because as I said, it’s a transfinite process. So at day omega, that’s the first infinite day, we’re going to create a lot of new surreal numbers. So every real number will be born at that stage, because every real number fills a gap in the previously born rational numbers that we had just talked about. It’s not all the rationals, because actually the rational numbers that are born at the finite stages are just the rationals whose denominator is a power of two, it turns out. Those are called the dyadic rationals.

Joel David Hamkins(02:51:57) So the real numbers are all born on day omega, but also some other numbers are born on day omega. Namely, the ordinal omega itself is the firstborn number that’s bigger than all those finite numbers, and minus omega is the firstborn number that’s less than all those finite numbers. But also, we have the number epsilon, which is the firstborn number that’s strictly bigger than zero and strictly less than all the positive rational numbers. So that’s going to be an infinitesimal number in that gap, and so on. On day omega plus one, we get more numbers, and then omega plus two and so on. And the numbers just keep coming forever. So, this is how you build the surreal number system.

Joel David Hamkins(02:52:39) And then it turns out you can define the arithmetic operations of addition and multiplication in a natural way that is engaging with this recursive definition. So we have sort of recursive definitions of plus and times for the surreal numbers. And it turns out you can prove that they make the surreal numbers into what’s called an ordered field. So they satisfy the field axioms, which means that you have distributivity and commutativity of addition and multiplication, and also you have reciprocals for every non-zero number. You can divide by the number. So you can add and multiply and divide and subtract. And furthermore, you can take square roots.

Joel David Hamkins(02:53:21) And furthermore, every odd degree polynomial has a root, which is true in the real numbers, because if you think about, say, a cubic or a fifth degree polynomial, then you know it’s going to cross the axis, because it has opposite behaviors on the two infinities, because it’s an odd degree polynomial. So on the positive side, it’s going to the positive infinity. On the negative side, it would be going to minus infinity. So it has to cross. So we know in the real numbers, every odd degree polynomial has a root. And that’s also true in the surreal numbers. So that makes it what’s called a real closed field which is a very nice mathematical theory. So it’s really quite interesting how we can find copies of all these other number systems inside the surreal numbers.

Lex Fridman(02:54:09) But the surreal numbers are fundamentally discontinuous as you’re worried about. What are the consequences of this?

Joel David Hamkins(02:54:14) Right. So the surreal numbers have a property that they form a non-standard model of the real field, which means that they provide a notion of infinitesimality that one can use to develop calculus on the grounds of Robinson’s non-standard theory that I had mentioned earlier. But they don’t have the least upper bound property for subcollections. There’s no set of surreal numbers, no non-trivial set of surreal numbers has at least upper bound, and there are no convergent sequences in the surreal numbers. And so for the sort of ordinary use in calculus based on limits and convergence, that method does not work in the surreal numbers at all. So that’s what I mean when I say the surreal numbers are fundamentally discontinuous. They have a fundamental discontinuity going on.

Joel David Hamkins(02:55:07) But you can still do calculus with them, because you have infinitesimals if you use these non-standard methods, the infinitesimal based methods to calculus. And people do that. I once organized a conference in New York, and we had John Conway as a speaker at that conference. And there was a question session, and someone asked him, I mean, it’s a bit of a rude question, I think, but they asked it and the question was, “What is your greatest disappointment in life?” I mean, I would never ask a question like that at a conference in a very public setting.

Joel David Hamkins(02:55:41) But Conway was extremely graceful and he answered by saying that, “The surreal numbers…” Not the numbers themselves, but the reception of the surreal numbers, because he had ambition that the surreal numbers would become a fundamental number system used throughout mathematics and science, because it was able to do nonstandard -set analysis, it was able to do calculus, it unified the ordinals and so on. And it’s such a unifying, amazing structure, beautiful structure with elegant proofs and sophisticated ideas all around it. And he was disappointed that it never really achieved that unifying status that he had the ambition for. And this, he mentioned as his greatest disappointment.

Lex Fridman(02:56:32) Yeah, Donald Knuth tried to celebrate it, but it never quite took hold.

Joel David Hamkins(02:56:36) So I don’t want to give the impression, though, that the surreal numbers are not widely studied, because there are thousands of people who are…

Lex Fridman(02:56:41) Sure

Joel David Hamkins(02:56:42) …studying it. In fact, Philip Ehrlich, who is one of the world experts on the surreal numbers, mentioned to me once that Conway was his own worst enemy with regard to that very issue because in the Conway style, everything is a game. And he treated the surreal numbers as a kind of plaything, a toy, and maybe that makes people not take it seriously. Although my view is that it is extremely serious and useful and profound, and I’ve been writing a whole series of essays on the surreal numbers for my Substack at Infinitely More. And I just find the whole subject so fascinating and beautiful. I mean, it’s true. I’m not applying it in engineering, which maybe was part of this Conway ambition.

Lex Fridman(02:57:30) And I just wanted to, before I forget, mention Conway turning everything into a game. It is a fascinating point that I didn’t quite think about, which I think the Game of Life is just an example of exploration of cellular automata. I think cellular automata is one of the most incredible, complicated, fascinating… It feels like an open door into a world we have not quite yet explored. And it’s such a beautiful illustration of that world, the Game of Life, but calling it a game… Maybe life balances it, because that’s your powerful word, but it’s not quite a game. It’s a fascinating invitation to an incredibly complicated and fascinating mathematical world.

Lex Fridman(02:58:09) I think every time I see cellular automata and the fact that we don’t quite have mathematical tools to make sense of that world, it fills me with awe. Speaking of a thousand years from now, it feels like that is a world we might make some progress on.

Joel David Hamkins(02:58:23) The Game of Life is a sort of playground for computably undecidable questions because, in fact, you can prove that the question of whether a given cell will ever become alive is computably undecidable. In other words…

Lex Fridman(02:58:39) Yeah

Joel David Hamkins(02:58:39) …given a configuration, and you ask, “Will this particular cell ever, you know, be alive—” …in the evolution?” And you can prove that that question is equivalent to the halting problem. It’s computably undecidable. It’s semi-decidable in the sense that if it will become alive, then you will know it at a finite stage because you could just run the Game of Life algorithm and let it run. And if it ever did come alive, you could say, “Yeah, it was alive.” But if you’ve run it for a thousand years and it hasn’t come alive yet, then you don’t necessarily seem to have any basis for saying, “No, it won’t ever come alive,” if the behavior was very complicated.

Joel David Hamkins(02:59:18) Maybe if you have a complete understanding of the evolution of the behavior, then you can say no, but you can prove you won’t always have that understanding— …precisely because the problem is equivalent to the halting problem.

Lex Fridman(02:59:28) And nevertheless, when you sit back and look and visualize the thing, some little mini cellular automata civilizations are born and die quickly, and some are very predictable and boring, but some have this rich, incredible complexity.

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See the full transcript and watch the full video episode for more. I shall periodically be posting more excerpts like this one here on Infinitely More—find them in the lex-fridman tag.

Read more about the surreal numbers in my series of essays in surreal-numbers tag, including the introductory essay The Surreal Numbers.

In particular, we shall have a purely order-theoretic account, the merge product α ▪ β, which I prefer to conceive as the principal semantic concept, although in mathematical practice this is less often given as the main definition; next a computational account I shall denote by α ⊗ β, based on the Cantor normal form, along with a closely related formal polynomial account α ⊛ β; after this, we shall have a definition of the natural product α ⊙ β by transfinite recursion; and finally, the multiplication of ordinals that arises in the surreal numbers α • β. Ultimately, we shall prove that all five notions are identical—they are different equivalent ways of looking at the same operation, the natural product, which in the end we shall often denote simply by αβ.

The argument is subtle, certainly not routine, and so I shall be glad to give a slow, careful presentation here. I am especially glad to do so because to my way of thinking, this is a core result about the natural product, but unfortunately, the full result is not commonly available in one place—one finds it piecemeal, stated and proved only partially and indeed it is often stated without any proof.

So let’s get into the fine details of what I regard as a fundamental illuminating result on the nature of the ordinals, regarding one of the most beautiful and natural operations on the ordinals, the natural product.

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Check it out at your favorite booksellers.

• Amazon

• Barnes and Noble

• MIT Press Bookstore

• Other options

From the preface:

Come, let us explore infinity! We shall visit all my favorite paradoxes and conundrums. The ancient puzzles, confounding or intractable, will yield at times to our analysis. And what a joy it is to experience those Aha! moments—a flash of clarity lights the way out of the labyrinth. But alas, having escaped one maze, we shall often find ourselves immediately lost in another—a new paradox with new questions to answer. The puzzles of infinity are endless riddles nestled within one another.

The Book of Infinity was the original motivation for me to begin my substack Infinitely More. When I first arrived a few years ago at the University of Notre Dame from Oxford, I was asked by my new department what course I would most want to teach. My answer was a new course on infinity that I had long dreamed about—what fun it would be to share my ideas and puzzles with enthusiastic students, tracing the concept from ancient times to contemporary issues. I set furiously to work preparing this book, a series of vignettes on infinity, and we offered the course. I serialized the chapters on Infinitely More as they were completed—see the section The Book of Infinity. I’ve since taught the course several more times, and with further polishing and editing, the book is finally completed.

400 pages and 26 chapters:

1. The Book of Numbers

2. The Sand Reckoner

3. Zeno’s Paradox

4. The Method of Exhaustion

5. Supertasks

6. The Infinite Coastline Paradox

7. The Paradox of Giants

8. The Paradox of the Largest Tweetable Number

9. Potential Versus Actual Infinity

10. Equinumerosity and Comparison of Size

11. What Is the Infinite?

12. Hilbert’s Grand Hotel

13. Uncountable Infinity

14. How to Count

15. Transfinite Recursive Constructions

16. Slaying the Hydra

17. The Continuum Hypothesis

18. Throwing Darts at the Real Line

19. The Orders of Infinity

20. The Surreal Numbers

21. The Axiom of Choice

22. Infinitary Hat Puzzles and the Aftermath

23. The Guessing-Box Puzzle

24. We Can Predict the Future

25. Infinite Liars

26. Common Knowledge

Here are a few snippets from the index, to give you an idea of what’s covered…

The book is packed with full-color mathematical figures—over 200 color figures, of my own design, which I produced in LaTeX using TikZ. Here are a few samples:

And many others! Each figure is woven into the text to help explain a mathematical or philosophical idea.

Meanwhile, I am serializing all my other books-in-progress here on Infinitely More—subscribe for full access to all my current work, including the surreal numbers, games, logic, philosophy of mathematics, and more.

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Notably, the natural sum and product on ordinals are both commutative operations—unlike the standard ordinal arithmetic—and so the natural semiring of ordinals is a commutative semiring. In fact, the natural sum and product operations on ordinals are the same operations that the ordinals exhibit in the surreal numbers, which makes the natural semiring of ordinals a subsemiring of the surreal numbers.

I shall describe several independent and self-standing approaches to the natural sum and product—we shall ultimately have five separate accounts of each operation, which proceed from and express different philosophical perspectives on how we should best undertake mathematical definitions with the ordinals. One account of the natural sum, for example, offers a purely order-theoretic structuralist account, while another can be seen as motivated by essentially computational concerns—how to compute the sum and product values—and still another account adopts in effect a proof-theoretic perspective by presenting a formal transfinite recursion. Ultimately, of course, we shall prove that the various alternative accounts of the natural sum and product are equivalent—they all ultimately define the same ordinal operations of the natural sum and product.

This is a happy situation, therefore, since to have multiple independent accounts of the same underlying mathematical idea is often valuable for mathematical insight. The different but ultimately equivalent approaches to the topic enrich our mathematical understanding by stretching our knowledge in different but fruitful directions. Different perspectives suggest different avenues of generalization, and some perspectives can be more clarifying than others depending on the specific case.

Let’s get into it.

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The first infinite number is ω, but one can always add 1 more.

The next simple limit is ω + ω, which is the same as ω · 2, and so one continues. Each new limit ordinal begins a new block of ordinals of length ω, a new era of infinity.

We proceed to the next limit ordinal ω · 3, starting yet another era of infinity, then ω · 4 after that, and indeed ω · n + k for every finite n and k:

In this way we have counted to ω². The ordinals encountered along the way take the form ω · n + k for finite n and k.

Counting to ω² is rather like counting to 100. When we count to 100 you might notice that within each decade—the teens, the twenties, the thirties, and so on—it is just like counting to 10 again. In counting to 100, which is 10², we thus count to 10 altogether 10 times. Similarly, when we count to ω², we count to ω altogether ω many times. We start with the finite numbers, the original copy of ω, and then proceed from ω to ω · 2, from ω · 2 to ω · 3, and so on. In counting up to ω², we thus encounter ω many eras, each of size ω, in effect counting to ω altogether ω many times. And just as the numbers up to 100 have two digits in base ten, with the form 10 · n + k, similarly the ordinals up to ω² have the form ω · n + k, which is two digits in base ω.

The ordinal ω² is the first compound limit ordinal—a limit ordinal that is a limit of limit ordinals since ω² is the limit of ω · n as n increases in ω. In other words, ω² is a limit ordinal, but there is no largest limit ordinal below it. A simple limit ordinal, in contrast, is a limit ordinal that is not a compound limit—all simple limits take the form α + ω for some ordinal α.

Counting to ω^ω and beyond

But I should truly like us to count much further. We essentially repeat the process of counting to ω² when counting from ω² to ω² · 2, then again when counting further to ω² · 3, and similarly through every successive ω² · n. With ω many repetitions, we thus count to ω² · ω, which is the ordinal ω³. By repeating that process ω many times, we reach ω⁴, and so on. Thus we are on our way to the local peak ω^ω, which is the supremum of ωⁿ for all finite numbers n.

Continuing further, if we count like this to ω^ω altogether ω many times, first to ω^ω · 2, then to ω^ω · 3, and so on, then we shall reach ω^ω · ω, which is the same as ω^ω+1. In light of the difficulty of reaching ω^ω in the first place, however, and having had to do that work ω many times to reach ω^ω+1, we might notice that it was a troublesome burden for us to increase the exponent merely by 1. And we shall have infinitely more such trouble again to reach ω^ω+2, and then still infinitely more trouble to reach ω^ω+3, and so on. Each increase of the exponent by 1 requires an additional infinite duplication of all the preceding difficult work to that juncture. And yet we shall not stop counting. With perseverance we shall reach ω^ω·2 and beyond—every tiny increase in the exponent is an achievement to be celebrated.

With stoical fortitude, we thus find our way to ω^ω·3 and then to ω^ω·4, on the way to

Eventually, exceeding that we shall arrive at

and then

and so on. We likely find ourselves exhausted at each new height of achievement. Nevertheless, we continue onward to

With enduring heroic dedication, we press on ever upward, successively scaling the towering further summits:

Each new step up with these finite-stack tetrations is a vast increase over the previous instance—remember how difficult it was to increase the exponent just by 1, but here we see huge steps up with vast exponential towers of increase. Nevertheless, with silent resolve and quiet determination we shall climb through these iterated exponential powers. The supremum of these finite-stack tetrations is a vast pinnacle, the ordinal known as ε₀.

Let us get started more seriously.

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The conversation turned at one point to the question of who has been the greatest mathematician of all time. I demurred a bit at the question—explaining that I don’t organize my thinking about mathematicians in such a ranked list, and find insight wherever it might arise, which isn’t always only from the greats—but I did eventually give an answer, which you can find out below. The question was an opportunity to talk about differing mathematical styles, including my own mathematical style, which has served me very well in my mathematical investigations.

Please enjoy this excerpt from our extended conversation. The transcript is below.

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Lex Fridman(03:12:59) Sorry to ask the ridiculous question, but who is the greatest mathematician of all time? Who are the possible candidates? Euler, Gauss, Newton, Ramanujan, Hilbert. We mentioned Gödel, Turing, if you throw him into the bucket.

Joel David Hamkins(03:13:14) So this is, I think, an incredibly difficult question to answer. Personally, I don’t really think this way about ranking mathematicians by greatness. Um…

Lex Fridman(03:13:28) So you don’t have, like… You know, some people have a Taylor Swift poster in their dorm room. You don’t have it.

Joel David Hamkins(03:13:33) I mean, if you forced me to pick someone, it would probably be Archimedes because…

Lex Fridman(03:13:37) Archimedes

Joel David Hamkins(03:13:37) …he had such incredible achievements in such an early era, which totally transcended the work of the other people in his era. But I also have the view that I want to learn mathematics and gain mathematical insight from whoever can provide it and wherever I can find it. And this isn’t always just coming from the greats. Sometimes the greats are doing things that are just first and not… You know, somebody else could have easily been first. So there’s a kind of luck aspect to it when you go back and look at the achievements. And because of this progress issue in mathematics that we talked about earlier, namely we really do understand things much better now than they used to.

Joel David Hamkins(03:14:22) And when you look back at the achievements that had been made, then maybe you can imagine thinking, “Well, somebody else could’ve had that insight also.” And maybe they would have… It’s already a known phenomenon that disparate mathematicians end up proving essentially similar results at approximately the same time. But, okay, the person who did it first is getting the credit and so on.

Lex Fridman(03:14:48) What do you make of that? Because I see that sometimes when mathematicians… This also applies in physics and science, where completely separately, discoveries are made…

Joel David Hamkins(03:14:58) Right. Yeah.

Lex Fridman(03:14:58) …maybe at a very similar time. What does that mean?

Joel David Hamkins(03:15:01) It’s relatively common. I mean, I think it’s like certain ideas are in the air and being thought about but not fully articulated, and so this is the nature of growth in knowledge.

Lex Fridman(03:15:13) Do you understand where ideas come from?

Joel David Hamkins(03:15:16) Not really.

Lex Fridman(03:15:17) I mean, what’s your own process when you’re thinking through a problem?

Joel David Hamkins(03:15:22) Yeah, that’s another difficult question. I suppose it has to do with… My mathematical style, my style as a mathematician, is that I don’t really like difficult mathematics. What I love is simple, clear, easy-to-understand arguments that prove a surprising result. That’s my favorite situation. And actually, the question of whether it’s a new result or not is somehow less important to me. And so that has to do with this question of the greats and so on, whoever does it first. Because I think, for example, if you prove a new result with a bad argument or a complicated argument, that’s great because you proved something new. But I still want to see the beautiful, simple, because that’s what I can understand.

Joel David Hamkins(03:16:16) Also, I’m kind of naturally skeptical about any complicated argument because it might be wrong. And… …If I can’t really understand it fully, like every single step all at once in my head, then I’m just worried maybe it’s wrong. And so these different styles, sometimes mathematicians get involved with these enormous research projects that involve huge numbers of working parts and… …Different technology coming together. I mean, mathematical technology, not physical technology.

Lex Fridman(03:16:48) And sometimes it actually involves now more and more something like the Lean programming language where some parts are automated, so you have this gigantic…

Joel David Hamkins(03:16:54) Yeah, yeah, I see. Well, that’s another issue because maybe those things are less subject to skepticism when it’s validated…

Lex Fridman(03:17:02) Sure

Joel David Hamkins(03:17:02) …by Lean. But I’m thinking about the case where the arguments are just extremely complicated, and so I sort of worry whether it’s right or not, whereas you know, I like the simple thing. So I tend to have often worked on things that are a little bit off the beaten path from what other people are working on from that point of view.

Lex Fridman(03:17:23) Your curiosity draws you towards simplicity.

Joel David Hamkins(03:17:25) Yeah. I want to work on the things that I can understand and that are simple. Luckily, I’ve found that I’ve been able to make contributions that other people seem to like, in this way, in this style. So I’ve been fortunate from that point of view. My process always, though, and I’ve recommended this always to my students, is just a kind of playful curiosity. So whenever I have…

Joel David Hamkins(03:17:55) Whenever there’s an idea or a topic then I just play around with it and change little things or understand a basic case and then make it more complicated or press things a little bit on this side or apply the idea to my favorite example that’s relevant, and see what happens, or you just play around with ideas, and this often leads to insights that then lead to more methods or more, then pretty soon you’re making progress on the problem. So this is basically my method, is I just fool around with the ideas until I can see a path through towards something interesting… …And then prove that, and that’s worked extremely well for me. So I’m pretty pleased with that method.

Lex Fridman(03:18:47) You do like thought experiments where you anthropomorphize like you mentioned?

Joel David Hamkins(03:18:51) Yeah, yeah. So this is a basic tool. I mean, I use this all the time. You imagine a set-theoretic model, a model of ZFC, as like a place where you’re living, and you might travel to distant lands by forcing. This is a kind of metaphor for what’s going on. Of course, the actual arguments aren’t anything like that because there’s not land and you’re not traveling and you’re not…

Lex Fridman(03:19:13) But you allow your mind to visualize that kind of thing-

Joel David Hamkins(03:19:15) Yeah

Lex Fridman(03:19:15) … in the natural real world.

Joel David Hamkins(03:19:16) And it helps you to understand. Particularly when there are parts of the argument that are in tension with one another, then you can imagine that people are fighting or something. And those kinds of metaphors, or you imagine it in terms of a game theoretic, you know, two players trying to win. So that’s kind of tension. And those kinds of metaphorical ways of understanding a mathematical problem often are extremely helpful in realizing, aha, the enemy is going to pick this thing to be like that because, you know, it makes it more continuous or whatever, and then we should do this other thing in order to… So it makes you realize mathematical strategies for finding the answer and proving the theorem that you want to prove because of the ideas that come out of that anthropomorphization.

Lex Fridman(03:20:01) What do you think of somebody like Andrew Wiles, who spent seven years grinding at one of the hardest problems in the history of mathematics? And maybe contrasting that a little bit with somebody who’s also brilliant, Terence Tao, who basically says if he hits a wall, he just switches to a different problem and he comes back and so on. So it’s less of a focused grind for many years without any guarantee that you’ll get there, which is what Andrew Wiles went through.

Joel David Hamkins(03:20:30) Right.

Lex Fridman(03:20:30) Maybe Grigori Perelman did the same.

Joel David Hamkins(03:20:32) I mean, Wiles proved an amazing theorem, Fermat’s Last Theorem result is incredible. This is a totally different style than my own practice, though, of working in isolation. For me, mathematics is often a kind of social activity. I have… I counted, I mean, it’s pushing towards a hundred collaborators, co-authors on various papers and so on. And, you know, if anybody has an idea they want to talk about with me, if I’m interested in it, then I’m going to want to collaborate with them and we might solve the problem and have a joint paper or whatever. You want to have a joint paper? Let me-

Lex Fridman(03:21:06) Yeah, exactly. Let’s go.

Joel David Hamkins(03:21:08) So my approach to making mathematical progress tends to involve working with other people quite a lot rather than just working on my…

Joel David Hamkins(03:21:17) …own, and I enjoy that aspect very much. So I, personally, I couldn’t ever do what Wiles did. Maybe I’m missing out. Maybe if I locked myself, you know, in the bedroom and just worked on whatever, then I would solve it. But I tend to think that no, actually, being on MathOverflow so much and I’ve gotten so many ideas, so many papers have grown out of the MathOverflow conversations and back and forth. Someone posts a question and I post an answer on part of it, and then someone else has an idea and it turns into a full solution, and then we have a three-way paper coming out of that. That’s happened many times. And so for me, I enjoy this kind of social aspect to it. And it’s not just the social part.

Joel David Hamkins(03:22:01) Rather, that’s the nature of mathematical investigation as I see it, is putting forth mathematical ideas to other people and they respond to it in a way that helps me learn, helps them learn, and I think that’s a very productive way of undertaking mathematics.

Lex Fridman(03:22:20) I think it’s when you work solo on mathematics, from my outsider perspective, it seems terrifyingly lonely. And because you’re, especially if you do stick to a single problem, especially if that problem has broken many brilliant mathematicians in the past, that you’re really putting all your chips in. And just the torment… …The rollercoaster of day to day. Because I imagine you have these moments of hopeful break, mini breakthroughs, and then you have to deal with the occasional realization that, no, it was not a breakthrough, and that disappointment.

Lex Fridman(03:23:00) And then you have to go, like, a weekly, maybe daily disappointment where you hit a wall, and you have no other person to brainstorm with. You have no other avenue to pursue. And it’s, I don’t know, the mental fortitude it takes to go through that. But everybody’s different. Some people are recluse and just really find solace in that lone grind. I have to ask about Grisha Grigori Perelman. What do you think of him famously declining the Fields Medal and the Millennial Prize? So he stated, “I’m not interested in money or fame. The prize is completely irrelevant to me. If the proof is correct, then no other recognition is needed.” What do you think of him turning down the prize?

Joel David Hamkins(03:23:52) I guess what I think is that mathematics is full of a lot of different kinds of people. And my attitude is that, hey, it doesn’t matter. Maybe they have a good math idea, and so I want to talk to them and interact with them. And so I think the Perelman case is maybe an instance where, you know, he’s such a brilliant mind and he solved this extremely famous and difficult problem, and that is a huge achievement. But he also had these views about, you know, prizes and somehow, I don’t really fully understand why he would turn it down.

Lex Fridman(03:24:33) I do think I have a similar thing, just observing Olympic athletes that are, in many cases, don’t get paid very much, and they nevertheless dedicate their entire lives for the pursuit… … Of the gold medal. I think his case is a reminder that some of the greatest mathematicians, some of the greatest scientists and human beings do the thing they do, take on these problems for the love of it, not for the prizes or the money or any of that. Now, as you’re saying, if the money comes, you could use it for stuff. If the prizes come, and the fame, and so on, that might be useful. But the reason fundamentally the greats do it is because of the art itself.

Joel David Hamkins(03:25:13) Sure, I totally agree with that. I mean, I share the view. That’s, you know, that’s why I’m a mathematician is because I find the questions so compelling and I’ve spent my whole life thinking about these problems. But, you know, but like if I won an award…

Lex Fridman(03:25:32) Yeah, it’s great. It’s great. I mean, I’m pretty sure you don’t contribute to MathOverflow for the wealth and the power. That you gain. I mean, it’s, yeah, genuine curiosity.

Joel David Hamkins(03:25:46) Well, you asked who the greatest mathematician is, and of course if we want to be truly objective about it, we would need a kind of an objective criteria…

Lex Fridman(03:25:55) Criteria, yeah.

Joel David Hamkins(03:25:55) …about how to evaluate the relative, you know, strength and the reputation of various mathematicians. And so, of course, we should use MathOverflow score… …Because…

Lex Fridman(03:26:06) That you’re definitively… I mean, nobody’s objectively the greatest mathematician of all time.

Joel David Hamkins(03:26:10) Yes, that’s true. I’ve also argued that tenure and promotion decisions should be based…

Lex Fridman(03:26:15) Based on MathOverflow.

Joel David Hamkins(03:26:16) …Yeah. So my daughter introduced me to her boyfriend. …And told me that she had a boyfriend. And I, um…

Lex Fridman(03:26:25) Asked him what his MathOverflow…

Joel David Hamkins(03:26:26) I wanted to know, first of all, what is his chess rating, and secondly, what is his MathOverflow score?

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See the full transcript and watch the full video episode for more. I shall periodically be posting more excerpts like this one here on Infinitely More—find them in the lex-fridman tag.

Let’s get into it, learning how to do ordinal arithmetic easily in the Cantor normal form. There are some surprising tricks that allow whole parts of the expressions to simplify away to nothing—they just disappear, leaving only the correct simplified expression behind.

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If you add two finite numbers together, then the resulting sum, of course, remains finite. Thus, the finite numbers are closed under addition. We may view this phenomenon as expressing a closure property of the ordinal ω itself, the first infinite ordinal. Namely, ω is closed under addition—the sum of any two numbers smaller than ω, that is, the sum of any two finite numbers, has a result that is still below ω. This is what it means to say that the ordinal ω is additively indecomposable.

Which other ordinals have this closure property? For example, what is the next additively indecomposable ordinal after ω?

Think about it…

Well, if an ordinal above ω is closed under addition, it would have to be bigger than ω + ω, and bigger than ω + ω + ω, and so forth. It would have to be bigger than ω · n for every finite number n. So it will be at least ω², the first compound limit ordinal.

Meanwhile, we can observe that ω² itself is closed under addition. Namely, if ordinals α, β are both below ω², then they are both below some ω · n for some large enough finite n, and consequently, we may bound the sum α + β as follows:

In other words, if α and β are smaller than ω², then α + β also is smaller than ω²—so the ordinal ω² is closed under addition. That is, ω² is additively indecomposable.

Additively Indecomposable Ordinals

In the general case, we define that an ordinal λ is additively indecomposable if every finite sum of ordinals below λ remains below λ. The ordinal ω, for example, is additively indecomposable, since the sum of finitely many finite numbers is finite, and we just observed above that ω² is additively indecomposable. The number 1 also is additively indecomposable, since every finite sum of ordinals below 1 adds up to 0, which remains less than 1.

What about 0 itself? Is 0 additively indecomposable? One might be inclined to say that 0 is additively indecomposable in a vacuous manner, since there are no smaller ordinals and therefore no finite sums of smaller ordinals. But this is not actually quite right according to the letter of the definition in light of the empty sum, which after all is a finite sum of ordinals, vacuously all less than 0, but the empty sum has value 0, which is not less than 0. On this technicality, therefore, 0 does not officially count as additively indecomposable. But actually, this is the answer we shall ultimately want, for this outcome makes for a smoother theory overall. The situation is similar to the question in number theory of whether the number 1 counts as prime. Sure, the only factors are 1 and itself, and yet mathematicians have agreed that we should not count 1 as prime. The number 1 after all is the value of the empty product, which is vacuously a product of smaller numbers. So 1 can be factored as a product of smaller numbers, the empty product, just as 0 is the value of a finite sum of smaller numbers, the empty sum.

Meanwhile, this consideration about the empty sum can also simply be absorbed into the statement by defining equivalently that an ordinal λ is additively indecomposable if 0 < λ and α + β < λ whenever α, β < λ. The stipulation that 0 < λ handles the empty sum, and the other finite sums are generated from ordinals below λ by adding extra terms one at a time. And so one often sees this latter definition, avoiding any need to consider the empty sum.

The additive indecomposability of an ordinal λ turns out to be equivalent to λ being additively irreducible, meaning that it cannot be expressed as a finite sum of smaller ordinals. Since 0 is the empty sum, an ordinal λ is additively irreducible if and only if 0 < λ and it is impossible that λ = α + β for some α, β < λ. (The reader is asked to prove the equivalence of additive indecomposability and irreducibility in the questions for further thought.)

Characterizing additive indecomposability

Which ordinals exactly are additively indecomposable? Can we characterize them? And what about multiplicative indecomposability? Is that the same as multiplicative irreducibility? And what about exponential indecomposability?

Think about it...

Let’s get into it…

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In this series of essays—you can find them in the ordinals tag—I shall cover all the basics of ordinal arithmetic, starting with the standard addition, multiplication, and exponentiation operations, but eventually getting to indecomposable ordinals, irreducible ordinals, Cantor normal form, binary ordinal representation, and more. Those who are new to the ordinals might want to start with How to Count.

A foundation for what is coming

I plan to lay down a solid foundation on these ordinal matters, with the aim in subsequent posts to grow the discussion into several deeper matters beyond, which require mastery over these concepts.

In particular, we shall continue on with my essays on the surreal numbers (in the surreal numbers tag), since our further work with that requires grounding in the ordinals. To foreshadow the coming topics, I shall subsequently introduce and investigate the so-called natural ordinal arithmetic (also known as the Hessenberg operations), which unlike the standard classical ordinal arithmetic are commutative operations. With the natural operations, the ordinals form a commutative semiring, the natural semiring of ordinals. The relevance of the natural ordinal arithmetic for the larger project of this book is that these are the same operations the ordinals experience in the surreal number field. After this, I shall introduce and develop the theory of what I call the natural ring of ordinals ⟨Ord⟩, which is the commutative ring generated by the ordinals under these operations, in which you can form such numbers as ω³ · 5 - ω² + ω - 7. This is precisely the subring generated by the ordinal numbers in the surreal field. How does the natural ring of ordinals compare with the Omnific integers? We saw that √2 is rational in Oz, for example, but what is the situation in the natural ring of ordinals ⟨Ord⟩? Do we have unique factorization in ⟨Ord⟩? Those are the questions at which we shall aim in the coming posts.

But first, in this essay, we shall put down a proper reliable foundation for the standard order-theoretic operations and basic theory of the ordinals. Everything coming later will build upon this.

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Namely, in this final essay I should like to discuss and defend what I see as an underlying potentialist nature to ultrafinitism. Specifically, I propose that we may fruitfully view the ultrafinitist perspective in a potentialist light, which will help illuminate its philosophical commitments, whilst also enabling a formal treatment of various ultrafinitist theories. What is more, I believe that the potentialist perspective brings to light certain fundamental issues on the nature of mathematical existence on which differing ultrafinitists might disagree, but which are most naturally discussed and adjudicated in a potentialist setting.

At the 2025 conference on ultrafinitism at Columbia University, Sam Buss mentioned that Ed Nelson had expressed ideas having a certain affinity with a potentialist outlook, in particular, the idea that things become true in arithmetic as you develop the theory—perhaps the twin primes conjecture could become true or the negation, depending on how the theory develops.

From the point of view of this essay, however, my entry into potentialism arises instead from a semantical perspective regarding the models of arithmetic, including models of the weak or the ultrafinitist theories. In a previous post, we saw, for example, how every model of finite arithmetic M ⊨ FA extends to taller models M⁺ and M⁺⁺ and so forth, with which it is bi-interpretable, and in another post, we saw that M extends ultimately to the limit model M* ⊨IΔ₀ of bounded induction. My view on this is to take it directly as a form of potentialism. Even if a finite-arithmetic ultrafinitist does not agree with the M* limit construction, nevertheless it seems to be in accordance with ultrafinitism to allow the move from M to M⁺, and this kind of move already leads exactly to the potentialist picture for arithmetic that I would like to paint.

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At one point, we talked about an instance of my advice to young mathematicians on the value of anthropomorphization as a means to achieve mathematical insight, applied in the case of Cantor’s argument on uncountable infinities and the Russell paradox. Please enjoy the excerpt—the transcript appears below.

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Lex Fridman(00:49:23) You mentioned to me offline we were talking about Russell’s paradox and that there’s another kind of anthropomorphizable proof of uncountability. I was wondering if you can lay that out.

Joel David Hamkins(00:49:41) Oh yeah, sure. Absolutely.

Lex Fridman(00:49:42) Both Russell’s paradox and the proof.

Joel David Hamkins(00:49:44) Right. So we talked about Cantor’s proof that the real numbers, the set of real numbers is an uncountable infinity, it’s a strictly larger infinity than the natural numbers. But Cantor actually proved a much more general fact, namely that for any set whatsoever, the power set of that set is a strictly larger set. So the power set is the set containing all the subsets of the original set. So if you have a set and you look at the collection of all of its subsets, then Cantor proved that this is a bigger set. They’re not equinumerous. Of course, there’s always at least as many subsets as elements because for any element, you can make the singleton subset that has only that guy as a member, right? So there’s always at least as many subsets as elements.

Joel David Hamkins(00:50:36) But the question is whether it’s strictly more or not. And so Cantor reasoned like this. It’s very simple. It’s a kind of distilling the abstract diagonalization idea without being encumbered by the complexity of the real numbers. So we have a set X and we’re looking at all of its subsets. That’s the power set of X. Suppose that X and the power set of X have the same size, suppose towards contradiction, they have the same size. So that means we can associate to every individual of X a subset. And so now let me define a new set. I mean, another set, I’m going to define it. Let’s call it D. And D is the subset of X that contains all the individuals that are not in their set.

Joel David Hamkins(00:51:28) Every individual was associated with a subset of X, and I’m looking at the individuals that are not in their set. Maybe nobody’s like that. Maybe there’s no element of X that’s like that, or maybe they’re all like that, or maybe some of them are and some of them aren’t. It doesn’t really matter for the argument. I defined a subset D consisting of the individuals that are not in the set that’s attached to them, but that’s a perfectly good subset. And so because of the equinumerosity, it would have to be attached to a particular individual, you know? And- Let’s call that person, it should be a name starting with D, so Diana.

Joel David Hamkins(00:52:10) And now we ask, is Diana an element of D or not? But if Diana is an element of D, then she is in her set. So she shouldn’t be because the set D was the set of individuals that are not in their set. So if Diana is in D, then she shouldn’t be. But if she isn’t in D, then she wouldn’t be in her set. And so she should be in D. That’s a contradiction. So therefore, the number of subsets is always greater than the number of elements for any set. And the anthropomorphizing idea is the following. I’d like to talk about it this way. For any collection of people, you can form more committees from them than there are people, even if you have infinitely many people.

Joel David Hamkins(00:53:03) Suppose you have an infinite set of people, and what’s a committee? Well, a committee is just a list of who’s on the committee basically, the members of the committee. So there’s all the two-person committees and there’s all the one-person committees and there’s the universal, the worst committee, the one that everyone is on. Okay. The best committee is the empty committee. With no members and never meets and so on. Or is the empty committee meeting all the time? I’m not sure.

Lex Fridman(00:53:29) Yeah. That’s… wow, that’s a profound question. And does a committee with just one member meet also?

Joel David Hamkins(00:53:35) Yeah. Maybe it’s always in session. I don’t know. So the claim is that there are more committees than people. Okay. Suppose not. Well, then we could make an association between the people and the committees. So we would have a kind of… every committee could be named after a person in a one-to-one way. And I’m not saying that the person is on the committee that’s named after them or not on it, whatever. Maybe sometimes that happens, sometimes it doesn’t. I don’t know. It doesn’t matter. But let’s form what I call committee D, which consists of all the people that are not on the committee that’s named after them.

Joel David Hamkins(00:54:16) Okay. Maybe that’s everyone, maybe it’s no one, maybe it’s half the people. It doesn’t matter. That’s a committee, it’s a set of people. And so it has to be named after someone. Let’s call that person Daniella. So now we ask, is Daniella on the committee that’s named after her? Well, if she is, then she shouldn’t be because it was the committee of people who aren’t on their own committee. And if she isn’t, then she should be. So again, it’s a contradiction. So when I was teaching at Oxford, one of my students came up with the following different anthropomorphization of Cantor’s argument. Let’s consider all possible fruit salads. We have a given collection of fruits.

Joel David Hamkins(00:55:07) You know, apples and oranges and grapes, whatever. And a fruit salad consists of some collection of those fruits. So there’s the banana, pear, grape salad and so on. There’s a lot of different kinds of salad. Every set of fruits makes a salad, a fruit salad. Okay… And we want to prove that for any collection of fruits, even if there are infinitely many different kinds of fruit, for any collection of fruits, there are more possible fruit salads than there are fruits. So if not, then you can put a one-to-one correspondence between the fruits and the fruit salads, so you could name every fruit salad after a fruit. That fruit might not be in that salad, it doesn’t matter. We’re just… it’s a naming, a one-to-one correspondence.

Joel David Hamkins(00:55:53) And then, of course, we form the diagonal salad, which consists… Of all the fruits that are not in the salad that’s named after them. And that’s a perfectly good salad. It might be a kind of diet salad, if it was the empty salad, or it might be the universal salad…

Joel David Hamkins(00:56:12) which had all fruits in it, if all the fruits were in it. Or it might have just some and not all. So that diagonal salad would have to be named after some fruit. So let’s suppose it’s named after durian, meaning that it was associated with durian in the one-to-one correspondence. And then we ask, well, is durian in the salad that it’s named after? And if it is, then it shouldn’t be. And if it isn’t, then it should be. And so it’s, again, the same contradiction. So all of those arguments are just the same as Cantor’s proof that the power set of any set is bigger than the set.

Joel David Hamkins(00:56:48) And this is exactly the same logic that comes up in Russell’s paradox, because Russell is arguing that the class of all sets can’t be a set because if it were, then we could form the set of all sets that are not elements of themselves. So basically, what Russell is proving is that there are more collections of sets than elements. Because we can form the diagonal class, you know, the class of all sets that are not elements of themselves. If that were a set, then it would be an element of itself if and only if it was not an element of itself. It’s exactly the same logic in all four of those arguments. So there can’t be a class of all sets, because if there were, then there would have to be a class of all sets that aren’t elements of themselves.

Joel David Hamkins(00:57:40) But that set would be an element of itself if and only if it’s not an element of itself, which is a contradiction. So this is the essence of the Russell paradox. I don’t call it the Russell paradox. Actually, when I teach it, I call it Russell’s theorem. There’s no universal set. And it’s not really confusing anymore. At the time, it was very confusing, but now we’ve absorbed this nature of set theory into our fundamental understanding of how sets are, and it’s not confusing anymore. I mean, the history is fascinating though, about the Russell paradox, because before that time, Frege was working on his monumental work undertaking, implementing the philosophy of logicism, which is the attempt to reduce all of mathematics to logic.

Joel David Hamkins(00:58:30) So Frege wanted to give an account of all of mathematics in terms of logical notions, and he was writing this monumental work and had formulated his basic principles. And those principles happened to imply that for any property whatsoever, you could form the set of objects with that property. This is known as the general comprehension principle. And he was appealing to the principles that support that axiom throughout his work. I mean, it was really… It wasn’t just an incidental thing, he was really using this principle.

Joel David Hamkins(00:59:11) And Russell wrote him a letter when he observed the work in progress, that there was this problem, because if you accept the principle that for any property whatsoever you can make a set of objects with that property, then you could form the set of all sets that are not members of themselves. That’s just an instance of the general comprehension principle. And… But the set of all sets that aren’t elements of themselves can’t be a set, because if it were, then it would be an element of itself if and only if it’s not a member of itself, and that’s a contradiction. And so Russell wrote this letter to Frege, and it was just at the moment when Frege was finishing his work. It was already at the publishers and, you know, in press basically. But it’s completely devastating.

Joel David Hamkins(00:59:58) I mean, it must have been such a horrible situation for Frege to be placed in, because he’s finished this monumental work, you know, years of his life dedicated to this, and Russell finds this basically one-line proof of a contradiction in the fundamental principles of the thesis that completely destroys the whole system. And Frege had put in the appendix of his work a response to Russell’s letter in which he explained what happened, and he wrote very gracefully, “Hardly anything more unwelcome can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished. This is the position into which I was put by a letter from Mr.

Joel David Hamkins(01:00:46) Bertrand Russell as the printing of this volume was nearing completion.” And then he goes on to explain the matter, it concerns his basic law five and so on.

Lex Fridman(01:00:54) It’s heartbreaking. I mean, there’s nothing more traumatic to a person who dreams of constructing mathematics all from logic, to get a very clean, simple contradiction. I mean, that’s just…

Joel David Hamkins(01:01:08) You devote your life to… This work, and then it’s shown to be contradictory, and that must have been heartbreaking.

Lex Fridman(01:01:16) What do you think about the Frege project, the philosophy of logic, the dream of the power of logic… To construct a mathematical universe?

Joel David Hamkins(01:01:24) So, of course, the project of logicism did not die with Frege, and it was continued, and, you know, there’s a whole movement, the neologicists and so on, in contemporary times even. But my view of the matter is that really, we should view the main goals of logicism are basically completely fulfilled in the rise of set-theoretic foundationalism. I mean, when you view ZFC as the foundation of mathematics, and in my view, the principles of ZFC are fundamentally logical in character, including the axiom of choice, as I mentioned, as a principle of logic. This is a highly disputed point of view, though, because a lot of people take even the axiom of infinity as mathematical, inherently mathematical and not logical and so on.

Joel David Hamkins(01:02:14) But I think if you adopt the view that the principles of ZFC have to do with the principles of abstract, you know, set formation, which is fundamentally logical in character, then it’s complete success for logicism. So the fact that set theory is able to serve as a foundation means that mathematics can be founded on logic.

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See the full transcript and watch the full video episode for more. I shall periodically be posting more excerpts like this one here on Infinitely More—find them in the lex-fridman tag.

We mentioned that some forms of ultrafinitism assert a realm of feasibility for the natural numbers, a realm which is closed under the successor operation, as well as addition and multiplication, but it is not closed exponentiation. Another totally different approach to ultrafinitism, meanwhile, posits explicitly the existence of a largest natural number. Although these two perspectives may seem initially to be completely at odds, I should like to explain in this essay how the model-theoretic semantics of these two positions are nevertheless intricately intertwined. Indeed, when viewed from a potentialist perspective, the two ultrafinitist positions come tightly together, sharing their essential potentialist ontological commitments. Are you intrigued?

The two theories I am talking about are the theory of finite arithmetic FA, which posits the existence of a largest number, and the theory of bounded induction IΔ₀, which proves the totality of addition and multiplication but not exponentiation.

The theorem I have in mind shows that these two theories are tightly connected in their model theory. Namely, we had observed previously that every model of IΔ₀ has all its truncations being models of FA. What I should like to prove in this essay is the converse—every model of finite arithmetic arises as the truncation of a model of bounded induction. Indeed, every model M of finite arithmetic has a unique minimal such extension M⁺ in which all the additive and multiplicative arithmetic of M becomes totally defined and deterministic. Thus, the models of FA and the models of IΔ₀ arrive together in their semantics—whenever you have a model M of FA, there is a unique smallest model M⁺ of IΔ₀ of which it arises as a truncation M = M⁺ ↾ n, and conversely, every truncation of a model of IΔ₀ has all its truncations being models of FA.

Let’s get into it.

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Some forms of ultrafinitism posit a class of feasible numbers and then assert that the number 0 is feasible; that the class of feasible numbers are closed under the successor operation n ↦ n + 1; and also that 2¹⁰⁰ is not feasible. Proponents then take pains to design the accompanying logical apparatus, of course, so as to prevent what would otherwise be the inevitable inconsistency by blocking attempts to carry out an inductive proof for that many steps. In the previous essay, I argued that these logical maneuvers amount in effect to implementing a measure of ultrafinitism in the metatheory, seeking in effect to allow only feasible terms and feasible proofs.

Meanwhile, I also discussed another, totally different approach to ultrafinitism, an approach that is less often discussed, but which I find to be in strong accordance with core ultrafinitist ideas. Namely, this alternative view arises from the simple idea that there could be a largest number. We introduced and discussed the theory I call finite arithmetic, abbreviated FA, which axiomatizes an approach to arithmetic with a largest number. I should like to further explore this theory today. This theory is also known as PA^top.

We discussed last time the question whether FA is the same as the common theory of all the standard truncation models ℕ ↾ n—we proved that it is not. In fact, the theory consisting of all sentences true in all the standard truncation models ℕ ↾ n admits of no computable axiomatization, since from any such axiomatization one could solve the halting problem. So it is difficult even for us to write down that common truncation-model theory, whereas FA consists of a simple list of axioms.

We also mentioned last time the ad hoc criticism, which objects to FA on the grounds that it cannot be our final, best account of the nature of arithmetic, in light of the deeply contingent, arbitrary nature of stopping at that particular largest number N. Why should the numbers stop sharply right at that point and not go on a little further? Why not define N + 1 somehow, as well as N · 2 and even N² somehow, and thereby provide a suitable, meaningful extension of the arithmetic operations up to larger numbers? It strikes many as absurd that our best, ultimate theory of arithmetic would posit the existence of a largest number.

In this essay, I shall aim to give mathematical legs to the ad hoc criticism by explaining the proof that every model of FA interprets a strictly taller model of this same theory. The fact of the matter is that inside any model of FA we can interpret a strictly taller model of FA in which the formerly largest number N now achieves its square N² or indeed N³ or much more. Indeed, it follows from this that in fact, all sums and products of members of M become fully defined in this taller model, even if they were not meaningful in the original model M. That is, all the undefined cases of sums and products in M become meaningful in the taller model. This picture therefore begins to reveal a fundamentally potentialist aspect to the nature of arithmetic, where the extent and nature of the arithmetic operations are gradually revealed via a possibility modality operator.

Ultimately, by iterating these interpretations to a limit, we shall prove next time that every model of FA arises via truncation from a model of bounded induction IΔ₀, a result due originally to Jeff Paris.

Let’s get into it.

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Is it an integer? We don’t know.

Merely denying the existence of all extremely large numbers, however, does not seem to make one an ultrafinitist. After all, other positions in the philosophy of mathematical ontology—I am thinking of certain forms of formalism, fictionalism, nominalism, and so forth—deny in a sense the existence of numbers altogether, but one would not ordinarily classify these views as automatically ultrafinitist. Ultrafinitism, rather, is specifically about a difference in the nature of existence of small versus large numbers. An ultrafinitist accepts the existence of the small or feasible numbers as unproblematic, yet denies the existence of very large numbers. Nevertheless, to be sure, there does seem to be a friendly affinity or overlap between those positions I mentioned and the ultrafinitist attitude toward very large numbers.

Harvey Friedman (2002, p. 4-5) raised the “draw the line” objection with ultrafinitist Alexander Yessenin-Volpin, concerning existence of 2¹⁰⁰.

I have seen some ultrafinitists go so far as to challenge the existence of 2¹⁰⁰ as a natural number, in the sense of there being a series of “points” of that length. There is the obvious “draw the line” objection, asking where in

do we stop having “Platonistic reality”? Here this ... is totally innocent, in that it can be easily be replaced by 100 items (names) separated by commas.

I raised just this objection with the (extreme) ultrafinitist Yessenin-Volpin during a lecture of his. He asked me to be more specific.

I then proceeded to start with 2¹ and asked him whether this is “real” or something to that effect. He virtually immediately said yes. Then I asked about 2², and he again said yes, but with a perceptible delay. Then 2³, and yes, but with more delay. This continued for a couple of more times, till it was obvious how he was handling this objection. Sure, he was prepared to always answer yes, but he was going to take 2¹⁰⁰ times as long to answer yes to 2¹⁰⁰ then he would to answering 2¹. There is no way that I could get very far with this.

Let’s discuss it.

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To recall the main concept, a set C in a topological space is said to be compact, if every open cover of C admits a finite subcover. That is, whenever 𝒰 is a collection of open sets and every point of C is an element of some U ∈ 𝒰—this is what it means for 𝒰 to be an open cover of C—then there is a finite subcover, meaning that there are finitely many sets in the collection U₀, ..., U_n ∈ 𝒰 that already cover C by themselves: C ⊆ U₀ ∪ ··· ∪ U_n.

For those who have not studied it before, perhaps this may seem to be a rather esoteric technical property for a set C to have—why should we care about this finite subcover property? Well, the answer is that mathematicians have found this notion again and again at the center of diverse mathematical phenomena. It comes up in the distinction between continuity and uniform continuity, the extreme value theorem, the property of topological normality, and many other situations. Compactness distills the essence of separate arguments on these topics, and it turns out that the compactness property itself directly implies the desirable topological features in these and many other cases. Thus, compactness is a unifying, explanatory notion. One may adopt a philosophical perspective on the matter, taking compactness to express an abstract form of finiteness—after all, every finite set is clearly compact. Like finiteness, the compactness of a space often enables a general transfer from local knowledge to the global counterpart. A typical example might be the fact that every continuous function on a compact set of real numbers is uniformly continuous—continuity is a local phenomenon, after all, and uniform continuity is global.

In light of this, it seems both natural and important to investigate the compactness phenomenon in the surreal line. How well do the compactness features of the real numbers carry over to the surreal numbers?

We shall find some surprises. To foreshadow, at first things will seem downright discouraging for any compactness phenomenon in the surreal numbers, since the naive approach to compactness in the surreal numbers will be an almost complete failure, yielding no nontrivial instances of compactness. And yet, upon reflection we shall hit upon an alternative, fruitful perspective, which will yield a robust collection of positive instances of compactness—all that would be expected or even hoped for. Indeed, more than expected! By the end of this essay I will present several surprising, even shocking instances of compactness in the surreal numbers.

Let’s get started.

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But what does it mean exactly to say that the real line is continuous? What does it mean for a topological space to be connected? And once we answer that question, I should like to know: Is the surreal line continuous in the same way? Is the surreal line No topologically connected?

Think about it...

The answer will be a pleasant philosophical surprise, for we shall see that the surreal line is disconnected with respect to one natural conception of connectedness; yet on another account—one giving great attention to the set/class distinction as it arises for the concept of open class—the surreal numbers are revealed as connected after all.

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Join me in this next instance of my ongoing series of essays on the surreal numbers. Today, we shall look at the ways in which the omnific integers don’t seem so very much like the integers after all. Several core features of the integers simply do not hold in the omnific integers.

You can find all the other surreal-number essays in the tag surreal-numbers, and readers who wish to review basic notions should look at The Surreal Numbers, which offers a gentle introduction to this amazing and fantastical structure.

In the previous post, we had introduced the omnific integers as providing an analogue of the integers for the surreal number system—the motivating idea was to view the omnific integers as sitting inside the surreal numbers in just the same way that the integers sit inside the real numbers. We had explored various senses in which the omnific integers Oz indeed play that role. In particular, the omnific integers Oz form an integer part of the surreal field No, which means that they are a discretely ordered subring for which every unit interval [x, x + 1) in the surreal numbers contains exactly one omnific integer.

However, I should like now to investigate several senses in which, regrettably, this analogy breaks down. The fact of the matter is that in several important respects, the omnific integers simply lack the properties that one might otherwise want to regard as core features of the integer ring as it is situated in the real field.

To give a taste of what is coming, we will show that the theory of prime numbers in the omnific integers is much less satisfactory than in the integers—there is, for example, no omnific analogue of the fundamental theorem of arithmetic, according to which every positive integer has a unique prime factorization. The first-order induction scheme fails for the nonnegative omnific integers. The process of placing fractions into lowest terms does not work properly with omnific integers. Indeed, to press this point sharply, we will prove specifically that the number √2 is actually rational with respect to the omnific integers Oz. And the number π is rational as well, it turns out, and you will find that there are many additional surprising instances of this phenomenon. In addition, Fermat’s last theorem, famously proved by Andrew Wiles to hold in the integers, is not true in the omnific integers—there are nontrivial omnific integer solutions to x³ + y³ = z³. Ultimately, therefore, we will see that many natural arithmetic principles that hold in the integers nevertheless fail in the omnific integers.

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This essay continues my ongoing series on the surreal numbers. This essay and all my other surreal numbers essays will appear in the tag surreal-numbers, with more content coming soon. Readers may wish to review basic notions in my introductory essay The Surreal Numbers, which offers a gentle introduction to this amazing and fantastical structure. For today, we continue the investigation by exploring the omnific integers!

Consider how the familiar integers ℤ sit inside the field of real numbers ℝ.

Every integer has an immediate successor integer just above it and an immediate predecessor just below, and so the integers are discretely ordered. Indeed, this is a discrete order with a uniform unit step size—we get to the next or previous integer always by adding or subtracting one. What is more, we may freely add, subtract, and multiply integers while remaining within the integer realm, and so they constitute what is called a discretely ordered subring of the real field. Furthermore, the integers span the entire real line, in the sense that every unit interval [x, x + 1] in the real numbers contains an integer. The integers thus form an integer-part of the real numbers, the canonical instance of this concept.

If we consider the surreal number system as a colossal analogue of the real numbers, then it would seem very natural to inquire whether we can find a corresponding colossal analogue of the integers inside the surreal number system. Can we find an integer part of the surreal numbers? We would seek a discretely ordered subring of the surreal numbers that furthermore spans the surreal line in the same way that the integers sit inside the real numbers

Interlude

Yes, indeed we can. Let us discover together the class of Omnific integers, a vast yet discretely ordered subring Oz of the surreal numbers No, closed under addition, subtraction, and multiplication, and with every omnific integer having a successor and predecessor in the omnific integers, each at unit distance from the next. What is more, the omnific integers fully span the surreal line—every unit interval in the surreal numbers contains an omnific integer, and thus they form an integer part of the surreal field. In light of these features, the omnific integers can be viewed as a surreal-number analogue to the integers sitting in the real field.

Let us explore the nature and basic theory of the omnific integers Oz. Eventually, we shall also happen to uncover several subtle failures of this analogy.

I intend to provide several fundamentally different constructions and characterizations of the beautifully crystalline structure of the omnific integers Oz. One often gains insight into a mathematical idea, after all, by investigating it from diverse perspectives, thereby often revealing fundamentally different aspects of it, which can lead to a deeper understanding. So let us adopt this strategy with the omnific integers. We shall begin with a purely order-theoretic construction of the omnific integers, proceeding in a surreal-numbers-style transfinite gap-filling manner, thereby constructing the omnific integers as a universal saturated endless discrete linear order. By subsuming that construction into the surreal number gap-filling construction, we will arrive at a characterization of the omnific integers in terms of the nature of their surreal sign-sequence representations. Finally, we shall also provide the convenient Conway surreal numeral characterization of the omnific integers as those surreal numbers satisfying x = { x - 1 | x + 1 }. Ultimately, we shall prove that all these various accounts give rise to exactly the same class of numbers, the plenitudinous class Oz of the omnific integers.

Let us begin.

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Welcome to this first essay in what will be a series on the surreal numbers, exploring all manner of curious topological aspects of the surreal field. Today we begin by looking into the nature of the surreal ω × ω chessboard. How many squares are there? How many pieces do we need to set up the board?

For a gentle introduction to the surreal numbers, readers may want to revisit my introductory essay The surreal numbers. You can also explore my surreal-number essays in the surreal-numbers tag, which will eventually include the upcoming essays in this series. Please enjoy!

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Introduction

Consider the ω × ω square in the surreal plane, an abundant playing ground, a generous arena in the midst of the vast surreal field.

I propose to partition the edge segments [0,ω] of the square into surreal unit lengths and thereby aim to construct a chessboard on this playing field. Shall we have a game of surreal chess? Naturally I intend that we shall use full-size living chessmen—there will be dangerous armed characters and scheming courtesans roaming the chessboard amongst the furious mounted knights, their horses stomping and snorting hot steamy air. Let us play surreal chess on the surreal ω × ω chessboard!

We shall presently be calling upon our courts and armies in order to set up the chessboard. But before the game begins, we shall first have to answer a certain mathematical puzzle: how many chess pieces altogether do we require? How many pawns shall we call up, for example, if we aim to set them up as expected, respectively, on the second and penultimate ranks? For that matter, how many square tiles are there altogether on the surreal ω × ω chessboard?

Interlude

The answer is one that many people find surprising…

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In recent posts we have been exploring the infinite subway paradox, starting with an introduction, followed by a fuller range of paradox, and then the transfinite extension of the paradox into the countable ordinals.

In this essay, we extend the paradox to the uncountably infinite. This essay engages therefore with a few more serious set-theoretic ideas, finding in the infinite subway paradox a call for the notions of stationarity, the club filter, and Fodor’s lemma on regressive functions. Please enjoy!

So let us consider the fully uncountable extension of the transfinite subway line. The train starts at station 0 and proceeds to station 1, station 2, through all the finite stations, to station ω, station ω + 1, station ω + 2, and so forth, eventually reaching station ω², station ω³, station ω^ω, station ε₀, and so on far beyond through all the countable ordinals. The train finally concludes its trip at the terminal station of the first uncountable ordinal, station ω₁. Which patterns of embarkment and disembarkment up to ω₁ are possible?

In the previous essays we saw how to reach any given countable ordinal with a finitely extensible train car, capable of holding any finite number of passengers at one time (but never infinitely many), while still having a disembarkment at every station stop along the way. The citizens of Infinitopolis can therefore fulfill the infinite subway challenge to reach any given countable ordinal.

Can we reach the first uncountable ordinal ω₁ itself this way? That is, can we unify all the various particular solutions reaching the various countable ordinals with a single uncountable schedule of passenger itineraries that proceeds all the way to ω₁? We know how to reach any particular countable ordinal α, yes, each with a separately defined schedule aimed at reaching that particular ordinal, but the question I am asking is whether we can do so in a fully uniform manner, whether we can describe a single uncountable trip, a single schedule of passenger itineraries in the finite extensible train that proceeds through every countable ordinal station in one run, while still having a disembarkment at every station stop along the way.

Interlude

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