In this instance, we shall consider the classical dispute between two conceptions of infinity—potential infinity versus actual infinity.

• A transcript of this lecture is available.

• Find the lectures here on Infinitely More in the lectures-on-infinity tag.

• The lectures are appearing on YouTube.

• The whole lecture course is hosted at Ergo: Lectures on Infinity.

• This lecture is based on my essay Potential versus Actual Infinity.

• The essay appears in my new book, The Book of Infinity.

Nevertheless, from the inception of the concept there was a certain troubling philosophical puzzle at its core, a philosophical problem lying in wait to upset those plans. Left unresolved, this issue could ultimately have formed a dangerous pitfall, preventing the success of the method. The issue to which I refer is:

Why is the notion of ordinal definability itself definable?

The question might have seemed especially worrisome and indeed a positive explanation unlikely in light of the fact that we already know from Tarski that the underlying notion of definability itself is not definable. A central lesson of many of the major developments of twentieth-century metamathematics—the Löwenheim-Skolem theorem, the related Skolem paradox, the compactness theorem, the existence of nonstandard models—is that first-order logic is weaker than one may initially expect, and we are often generally unable to express natural higher-order metamathematical notions within a given first-order object theory such as set theory.

So how is it that we entitled to express and refer to ordinal definability in set theory but not definability? Indeed we are able to refer to it—the notion of ordinal definability is definable—but how?

There is something about allowing ordinal parameters, it turns out, that enables us to accommodate what would otherwise be a purely philosophical, metatheoretic notion in the underlying theory—amazingly, we are able to bring this notion from the metatheory into the object theory. In this manner, a philosophical notion becomes mathematical, becoming part of the subject matter.

Let me tell you all about it—I shall explain everything.

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Is infinity even or odd?

How shall we answer?

Think about it...

The child’s question may seem innocent, but actually there are diverse ways to take it, leading to a corresponding spectrum of answers. Perhaps it is natural initially to answer from the point of view of the least infinite ordinal ω with the standard ordinal arithmetic, for example, but even in this case there are two distinct concepts of even, depending on whether we imagine dividing into a sequence of pairs or cutting in half; we shall get still another answer if we should use the natural arithmetic; and we shall find further different answers when using different infinite ordinals; or when considering the question in the natural ring of ordinals; or in the omnific integers; we can answer with the cardinal numbers of set theory, either with the axiom of choice, or without—and it matters. Thus we shall discover a great diversity of answers for the child’s question, senses in which infinity is even, senses in which it is not, senses in which the answer depends on the particular infinity we have in mind, and senses in which the answer depends on our mathematical axiomatic foundations.

Let us explore!

Even and Odd in the Ordinals

Let us begin by considering the concept of even and odd in the transfinite ordinal numbers with the standard ordinal arithmetic.

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In this instance, we shall consider the paradox of the largest tweetable number. What is the largest number that you can tweet? (to use the old terminology for making a post on X) The question will lead us on a cosmic journey to some truly astounding numbers and beyond, eventually into an evaporating mist of confusion, of logical complexity and paradox, ultimately revealing a fundamental underlying issue of mathematical determinacy—does every mathematical question have a determinate answer?

• A transcript of this lecture is available.

• Find the lectures here on Infinitely More in the lectures-on-infinity tag.

• The lectures will also appear on YouTube.

• The whole lecture course is hosted at Ergo: Lectures on Infinity.

• Find other philosophy lecture courses at Ergo.org.

• This lecture is based on my essay The largest tweetable number.

• The essay also appears in my new book, The Book of Infinity.

In this third lecture, we shall explore the paradox of giants, showcasing Galileo’s argument that the traditional giants of folklore—taking human form but at much larger scale—are physically impossible. He argued on the basis of an understanding of how size scales differently in different dimensions. Similar ideas lead to the paradox of Gabriel’s horn, the painter’s paradox, paradoxical fractals, and to many further paradoxes of dimension. By the end of the lecture, we shall glimpse some genuinely troubling conundrums in high dimension.

• A transcript of this lecture is available.

• Find the lectures here on Infinitely More in the lectures-on-infinity tag.

• The lectures will appear on YouTube.

• The whole lecture course is hosted at Ergo: Lectures on Infinity.

• Find other philosophy lecture courses at Ergo.org.

• This lecture is based on my essay The Paradox of Giants.

• The essay also appears in my new book, The Book of Infinity.

In this second lecture, we explore the concept of supertask—a task involving infinitely many separate actions or steps. We shall play with Thomson’s lamp, turning it on and off infinitely in a finite duration of time, before encountering the dangerous Deal with the Devil. And then infinitely many billiard balls in a sack! Will the Glutton win in the Chocolatier’s game? Let’s find out.

• A transcript of this lecture is available.

• Find the lectures here on Infinitely More in the lectures-on-infinity tag.

• The lectures will appear on YouTube.

• The whole lecture course is hosted at Ergo: Lectures on Infinity.

• Find other philosophy lecture courses at Ergo.org.

• This lecture is based on my essay Supertasks.

• The essay also appears in my new book, The Book of Infinity.

In this first lecture, we introduce the ancient puzzle of Zeno’s paradox and follow the thread where it leads—to the very meaning of our number expressions, to infinite summations, to convergent and divergent series. We will discuss what I call the most contested equation in middle school, and by the end we shall reach the remarkable Riemann rearrangement theorem, concerning the malleable nature of infinite sums.

• A transcript of the lecture is available.

• Find the lectures here on Infinitely More in the lectures-on-infinity tag.

• The lectures will appear on YouTube.

• The whole lecture course is hosted at Ergo: Lectures on Infinity.

• Find other philosophy lecture courses at Ergo.org.

• This lecture is based on my essay Zeno’s Paradox.

• The essay also appears in my new book, The Book of Infinity.

These lectures are an exploration of all my favorite paradoxes and conundrums. We’ll get into Zeno’s paradox, supertasks, the paradox of giants, the paradox of the largest tweetable number, Galileo’s paradox, the dispute between potentialist and actualist conceptions of infinity, and much more. Eventually, we’ll dive into the details of Cantor’s discovery of uncountable infinity, the problem of the continuum hypothesis, and much more. Please enjoy!

The lectures are based on my new book, The Book of Infinity, MIT Press 2026.

I have been serializing the chapters here on Infinitely More as I wrote them—find them in the section The Book of Infinity.

The full lecture series is hosted at Ergo.org in the form of a lecture course Ergo: Lectures on Infinity, part of the broad collection of philosophy lecture courses they have assembled there. Highly recommended, and I am proud to be part of it.

The lectures will also appear on the Ergo YouTube channel, in collaboration with my YouTube channel.

I shall be releasing the videos here on Infinitely More over the coming weeks and months, with full transcripts—subscribe now for access to these and all my other Infinitely More content. I’ll be sending out the first lecture very soon: Zeno’s paradox and infinite sums.

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Which numbers arise in this field? Can we represent √2 or √ω this way? Which surreal numbers are algebraic or transcendental over the ordinals? Do the ordinals reveal new algebraic relations concerning e or π? How can we know?

We shall discuss all this and more in today’s installment, part of my series of essays on the ordinal numbers—find them in the ordinals tag.

Let’s get into it!

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when the exponent n is larger than 2. For example, there is no nontrivial solution in the integers of a³ + b³ = c³ and no nontrivial solution of a⁴ + b⁴ = c⁴. By nontrivial, we just mean that the numbers a, b, c are all nonzero, since we don’t want to count. 2³ + 0³ = 2³ as a counterexample instance.

The theorem is named for Pierre de Fermat, who in 1637 scribbled a note claiming the result in the margin of book, saying that he had found a truly wonderful proof, but alas, the margin was too small to contain it.

Mathematicians spent centuries since that time struggling to find the missing proof, or indeed any proof at all, always failing, until finally Andrew Wiles proved the theorem in 1994. Wiles’s argument uses sophisticated contemporary ideas—almost surely not what Fermat had in mind. Indeed, many mathematicians believe that Fermat was probably mistaken about having a proof of the general result in the first place.

I propose that we should consider the question of Fermat’s last theorem in the natural ring of ordinals. Namely, are there nonzero numbers a, b, and c in the natural ring of ordinals ⟨Ord⟩ that solve

for an integer exponent n > 2? What do you think?

Think about it...

Let’s get into it!

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We discussed a sweeping selection of topics—the nature of mathematical truth, proof, platonism, fictionalism, the roles of set theory in mathematics, pluralism in the foundations of mathematics, the multiverse view, the continuum hypothesis, the junk-theorem phenomenon, the dispute between classical logic and constructive mathematics, and much more.

I should confess that one of my hosts and I are known to disagree on a few core philosophical matters—there are some Twitter exchanges to prove it—and our conversation here did not shy away from these points of contention. In a friendly spirit of constructive philosophical exchange, there were probing questions, challenges, and witty counterpoints. What an enjoyable and fruitful meeting of the minds it was. On which side of these questions do you find yourself? Post your views in the comments.

Please enjoy!

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Subscribe for full access to Infinitely More, with regular posts on infinity, philosophy, and mathematics.

Learn more at these links:

• Deniz and Thorston’s About Logic YouTube channel, where you’ll find numerous interviews with prominent logicians, mathematicians, and philosophers. Highly recommended.

• Joel David Hamkins YouTube channel

• My book, Lectures on the Philosophy of Mathematics

• My new book, The Book of Infinity

• Joel David Hamkins, “How the continuum hypothesis could have been a fundamental axiom,” Journal for the Philosophy of Mathematics (2024), DOI:10.36253/jpm-2936, arxiv:2407.02463.

In the previous essay, The Natural Ring of Ordinals Has Prime Factorization, we proved that this ring is an integral domain and indeed it is a unique factorization domain—every number factors uniquely as a finite product of primes. Thus, the ordinal analogue of the fundamental theorem of arithmetic holds in the natural ring of ordinals. Find the whole essay series in the ordinals tag.

Today, I should like to discuss several features in this ring that regrettably do not work out as one might have hoped or expected.

• The concept of even goes awry. Not every odd number has the form 2a＋1.

• The alternating even/odd pattern fails in the natural ring of ordinals. There are consecutive odd numbers!

• Indeed, there are arbitrarily long chains of consecutive odd numbers.

• The Collatz conjecture fails badly in the natural ring of ordinals.

• There are consecutive infinite prime numbers.

• There are arbitrarily long intervals of consecutive prime numbers.

• In this sense, the prime pair conjecture holds in the natural ring of ordinals in a very strong formulation.

• Greatest common divisors exist, but they do not always fulfill Bézout’s identity, by which the GCD is represented as a linear combination:

• The natural ring of ordinals ⟨Ord⟩ is not a Euclidean domain—we cannot always divide with remainder.

• We therefore cannot reliably implement the Euclidean algorithm.

• Indeed, the natural ring of ordinals is not a principal ideal domain, nor is it a Noetherian ring (these concepts will be explained).

• The natural ring of ordinals is not an integer part of the surreal field. There are arbitrarily large intervals in the surreal field containing no member of the natural ring of ordinals.

Let’s get into it!

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A ring is a certain kind of algebraic structure generalizing the familiar arithmetic of the integers ⟨ℤ,+,·⟩ to a more general or abstract realm of number objects. In ring theory, we aim to leverage our understanding of the integers to these more abstract realms. And indeed the natural ring of ordinals generalizes many features of the integers to a context including the transfinite ordinals. Today we shall how prime factorization is manifested in the natural ring of ordinals, with an ordinal version of the fundamental theorem of arithmetic—every number factors uniquely as a finite product of primes.

We introduced the basic construction of ⟨Ord⟩ in my previous essay, The Natural Ring of Ordinals, where we saw how the numbers of this ring can be represented as formal ordinal differences α ─ β, taken with respect to the same-difference equivalence relation. We outlined several attractive algebraic features to which this leads. The natural ring of ordinals, for example, is a discretely ordered cancellative commutative ring with identity. We provided several normal forms for representing the numbers in this ring, such as the signed Cantor normal form

where the coefficients k_i are taken from the integers, including negative integers. We saw the closely related normal form based on finite-support formal polynomial expressions ∑_β ω^β · k_β and another normal form based on finite signed sums of distinct powers of 2. The natural ring of ordinals, it turns out, is isomorphic to the subring of the surreal numbers generated by the ordinals, and so one may legitimately imagine these numbers, if desired, as surreal numbers. Meanwhile, the direct construction is simple and proceeds independently of any need for the surreal field.

We ended the previous essay on a cliff-hanger with several tantalizing questions left open, which I shall presently begin to take up in this essay:

• Does ⟨Ord⟩ admit a robust concept of prime numbers?

• Does ⟨Ord⟩ admit a robust concept of even and odd?

• Does ⟨Ord⟩ admit greatest common divisors?

• Is √2 irrational with respect to ⟨Ord⟩?

• What about √ω?

I shall aim to provide a general algebraic analysis of the natural ring of ordinals, showing that it is an integral domain and indeed, a unique factorization domain, which will be the key to several of the questions above. Next time, however, we will see that the natural ring of ordinals is not a Euclidean domain, nor a principal ideal domain, nor a Noetherian ring.

Our analysis here will be based on the extremely fruitful structural observation that the natural ring of ordinals is isomorphic to a vast discretely ordered polynomial ring over the integers, arising in a presentation as a transfinite tower of rings extending endlessly upward with newly created indefinite variables x_α, one for each ordinal α, with each new generator x_α added on top, larger in the order than every polynomial using only earlier variables.

Once we do this, it follows on general ring-theoretic grounds that ⟨Ord⟩ is a unique factorization domain and therefore supports a robust theory of irreducibility and prime numbers, by which every number factors uniquely as a finite product of primes. This observation in turn is the key to several of the questions asked above.

Let’s get into it!

You can find my whole essay series on the ordinals in the ordinals tag.

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A ring is a certain kind of algebraic structure generalizing the familiar arithmetic of the integers ⟨ℤ,+,·⟩ to a more general or abstract realm of number objects, such as polynomials or matrices. In the natural ring of ordinals, for example, we shall find the ordinals as well as formal ordinal differences α ─ β. Ring theory is about leveraging our deep understanding of the integers to these more abstract objects and contexts, which can often retain many of the attractive features of integer arithmetic in a manner that fruitfully aids our thinking about them, often leading to insight. We can add and multiply polynomials, for example, and factor them; in certain contexts we can perform long division of polynomials with remainders in a manner that is exactly analogous to what we do in the integers with the Euclidean algorithm. In the natural ring of ordinals, we shall find a robust concept of prime number and indeed every number in this ring will factor uniquely as a product of primes—thus we shall discover an ordinal analogue of the fundamental theorem of arithmetic. This feature is completely lacking, in contrast, in the ring of omnific integers in the surreal numbers, which exhibits only weaker ring-theoretic features.

To be more specific about the definition, a ring consists of a realm of objects with accompanying concepts of addition and multiplication for them exhibiting certain regular features, namely, both addition and multiplication are associative; addition is commutative; there is an additive identity 0; every number has an additive inverse; and lastly, multiplication distributes over addition.

Examples of rings would include the ring of integers ⟨ℤ,+,·⟩, of course, with the usual arithmetic; but also modular arithmetic ℤ/nℤ with addition and multiplication modulo n; the ring ℤ[x] of polynomials in the indefinite variable x with integer coefficients; the polynomial rings ℤ[x, y, z,…] allowing additional variables; the various field structures, such as the rational field ℚ; the real field ℝ; the complex field ℂ; and the associated polynomial rings over these fields, such as ℚ[x, y, z,…]. These are all commutative rings, meaning that multiplication as well as addition is commutative, but there are also many noncommutative rings, such as the various matrix rings ℝ^2×2 or ℂ^n×n, and many others.

Meanwhile, the natural numbers ℕ with the usual addition and multiplication do not form a ring, since they lack additive inverses, but the natural numbers do form what is called a semiring, which drops that requirement. Similarly, the ordinals are not a ring, since they also lack additive inverses, although with the natural sum and product they do form a semiring. We shall ultimately expand the class of ordinals with ideal number objects that will represent the various possible ordinal differences α ─ β, thereby providing a presentation of the natural ring of ordinals, which has many attractive features.

I denote the natural ring of ordinals by ⟨Ord⟩ in order to suggest that it is generated by but not identical to the class of ordinals. The natural ring of ordinals, it turns out, is a discretely ordered commutative ring, just like the integers themselves, and it is moreover an integral domain, which means nonzero numbers never multiply to zero. Further, it will turn out that the natural ring of ordinals is a unique factorization domain, which means that we shall find in it a robust concept of prime number and every number will factor uniquely as a finite product of primes.

Perhaps the reader will be surprised to learn that the natural ring of ordinals can be realized isomorphically as a vast discretely ordered polynomial ring over the integers, arising in a presentation as a transfinite tower of rings extending endlessly upward with newly created indefinite variables x_α, one for each ordinal α, with each new generator x_α added on top, larger in the order than every polynomial using only earlier variables.

We shall also find a variety of useful normal forms for the elements of the natural ring of ordinals, such as the signed analogue of the Cantor normal form,

with the difference that we now allow the coefficients k_i to be arbitrary integers, including negative integers, in order to accommodate the ordinal differences. Numbers in the natural ring of ordinals can also be represented as a signed sum of distinct powers of 2:

with α_n > ··· > α₀, although this representation is not unique, in light of such examples as 8 - 1 = 4 + 2 + 1.

The natural ring of ordinals, it turns out, is the same as the ring generated by the ordinal numbers inside the surreal number field—but it is strictly contained within the omnific integers.

To mention a few intriguing features, we shall eventually prove that √2 is irrational in the natural ring of ordinals, just as it is with the classical Pythagorean result in the integers. Of course, this would be the expected result, at least until one recalls that this wasn’t true in the omnific integers, since we saw earlier that √2 is rational with respect to the omnific integers and indeed every surreal number is rational with respect to the omnific integers. The number 2 is prime in the natural ring of ordinals and every number factors uniquely into primes, which might seem initially to give a solid basis for the concepts of even and odd in the natural ring of ordinals. However, these concepts are regrettably a little less successful there than in the integers, for matters go somewhat awry in regard to the order—we lose the uniformly regular even/odd pattern and indeed, there will be arbitrary long intervals in the natural ring of ordinals having no even numbers at all. This phenomenon can be seen as an instance of the failure of the Euclidean algorithm in the natural ring of ordinals—⟨Ord⟩ is not a Euclidean domain, nor even a principal ideal domain, nor is it Noetherian. Although we shall have a concept of greatest common divisor in the natural ring of ordinals, nevertheless the ordinal analogue of Bézout’s identity will fail—the greatest common divisor of two numbers will not always arise as a linear combination of them.

Eventually we shall also perform the quotient field construction and thereby construct the natural field of ordinals, as well as its real algebraic closure. Let us aim to explore all these ordinal number rings and fields together.

Let’s get into it!

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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.