Games of chess are thus weirdly truncated—one has the winning goal not of actually capturing the enemy king, but instead of placing oneself into a position in which one would be able to capture the king on the next move, no matter how the opponent plays next, if contrary to fact the moves continuing from that juncture were to be allowed. But how strangely metatheoretic and anticlimactic this is. Instead of simply continuing the game to its natural conclusion, executing what would be the main dramatic event, the game instead ends abruptly, prematurely. The anxiously anticipated king capture never actually occurs on the board. Games of chess thus seem always to end one move early—the curtains close just before the climax.

Why do we play chess like this? Why don’t we play chess with the more natural goal of simply capturing the opposing king? In what I call natural chess, we abandon the rules about check and checkmate, adopting instead the goal of capturing the opposing king. Is this simplified version of chess game-theoretically equivalent to standard chess? Are there the same game outcomes from a given position?

Think about it...

Surprisingly, the answer is no. Although natural chess is almost equivalent to standard chess, nevertheless it is not quite fully equivalent, even when we make reasonable changes to other standard rules such as those involving stalemate.

To show the basic inequivalence of standard chess and natural chess, I shall exhibit three kinds of chess positions that distinguish them. Namely, (1) there are chess positions which are a win according to the rules of standard chess, but in which it would not be possible for the player to capture the king in natural chess; conversely, (2) there are chess positions that are a draw in standard chess, but a win in natural chess; and finally (3) there are chess positions that are an immediate win in standard chess, but losing in natural chess. Standard chess, therefore, is not exactly the same as the necessary capture of the opposing king.

Really? Let’s discuss it.

Penultimate chess

Before continuing with that main theme, however, let me first continue to highlight what I see as a slightly comical absurdity in our standard chess rules. I should like to mock the standard rules. The basic inspiration for the standard checkmate rules, of course, is the idea that we needn’t actually capture the enemy king, but rather it should suffice for winning to be in a position such that we would certainly be able to do so on the next move.

Why don’t we take this one further step removed? Consider the chess variation I call penultimate chess. In penultimate chess, players should announce checkmate-check, when they are in a position threaten checkmate on the next move—that is, the current move is not necessarily checkmate nor even check, but they are threatening to make checkmate on their next move, unless the opponent moves so as to obstruct this. If indeed there is no way for the opponent to block the checkmate threat, then we shall say that the first player has placed his opponent in checkmate-mate—they are sure to be able to create checkmate on the next move. Of course, to make a checkmate-mate move is to create a mate-in-1 position, with the opponent to move (so it was mate-in-2 before the checkmate-mate move).

And now the point is that in penultimate chess we shall take checkmate-mate as the winning condition. You don’t have to actually capture the king, nor even put yourself in a position in which you would be certainly able to capture the king, that is, checkmate—rather, you win penultimate chess by placing yourself in a position in which you are certainly able to put yourself in a position in which you would be certainly able to capture the king, that is, you win by placing your opponent in checkmate-mate. Just as standard chess ends at checkmate, one step before the actual king capture, penultimate chess ends at checkmate-mate, one step before checkmate, or two steps before the king capture. I would find it entirely in keeping with the spirit of the standard checkmate rules that in penultimate chess we should also make it illegal for a player to leave themselves exposed to checkmate-check (even if they are not in check), just as it is illegal in the standard chess rules to leave one’s king in check.

And why stop there? In the game of antepenultimate chess, one seeks not to capture the king explicitly, nor to create a checkmate position (in which one would certainly be able to capture the king), nor a checkmate-mate position (in which one would be certainly able to be certainly able to capture the king), but rather to create a checkmate-mate-mate position, in which one is certainly able on the next move to create a situation in which one is certainly able to create a position in which one is certainly able to capture the king on the following move. Of course, these are the mate-in-2 positions. If we think of checkmate as king-capture-in-1, then mate-in-1 is king-capture-in-2, and so one wins penultimate chess with king-capture-in-2, and you win antepenultimate chess with king-capture-in-3, or mate-in-2.

Wouldn’t this be ridiculous? Not much more ridiculous, I would say, than our current standard rules for checkmate, when compared with simply playing to capture the king.

I should remark that all these games are affected by analogues of the analysis showing the difference between standard chess and natural chess, and so actually standard penultimate chess will not be game-theoretically identical to natural penultimate chess, and similarly with antepenultimate chess. Similarly, in preantepenultimate chess the goal is to make checkmate-mate-mate-mate, or king-capture-in-4, or mate-in-3.

Let’s get into it!

(A quick note regarding the phrase “one move early” is that it refers to the usual chess manner of counting moves, where one counts the moves of only one player. This is the usage in the white-to-mate-in-3 terminology, where it means that white will make three moves (or fewer if black plays poorly), so the moves will generally proceed white/black/white/black/white, ending in checkmate on the third white move. This method of counting also appears in the common chess notation such as 1.e4 e5 2.Nf3 Nc6, where each move by a player is referred to as a half move in the game.)

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In this instance, we shall consider Galileo’s paradox of infinity and the concept of equinumerosity for infinite collections.

• 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 Equinumerosity and Comparison of Size.

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

This is the historical conundrum of ordinal definability, which I should like to consider in this essay.

Let’s get into it.

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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.

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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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