Paradox, Infinity, & The Foundations of Mathematics, interview with Robinson Erhardt, January 2023

This was an interview with Robinson Erhardt on Robinson’s Podcast, part of his series of interviews with various philosophers, including many philosophers of mathematics and more.

We had a wonderfully wide-ranging discussion about the philosophy of mathematics, the philosophy of set theory, pluralism, and many other topics. The main focus was the topic of infinity, following selections from my new book, The Book of Infinity, currently being serialized on my substack,, with discussion of Zeno’s paradox, the Chocolatier’s Game, Hilbert’s Grand Hotel and more.

Robinson compiled the following outline with links to special parts of the interview:

  • 00:00 Introduction
  • 2:52 Is Joel a Mathematician or a Philosopher?
  • 6:13 The Philosophical Influence of Hugh Woodin
  • 10:29 The Intersection of Set Theory and Philosophy of Math
  • 16:29 Serializing the Book of the Infinite
  • 20:05 Zeno of Elea, Continuity, and Geometric Series
  • 39:39 Infinite Games and the Chocolatier
  • 53:35 Hilbert’s Hotel
  • 1:10:26 Cantor’s Theorem
  • 1:31:37 The Continuum Hypothesis
  • 1:43:02 The Set-Theoretic Multiverse
  • 2:00:25 Berry’s Paradox and Large Numbers
  • 2:16:15 Skolem’s Paradox and Indescribable Numbers
  • 2:28:41 Pascal’s Wager and Reasoning Around Remote Events
  • 2:49:35 MathOverflow
  • 3:04:40 Joel’s Impeccable Fashion Sense

Read the book here:

Infinity, University of Notre Dame, Spring 2023


Philosophy 20607 01 (32582)

University of Notre Dame                                                                              Spring 2023

Instructor: Joel David Hamkins, O’Hara Professor of Philosophy and Mathematics
3:30-4:45 Tuesdays + Thursdays, DeBartolo Hall 208

Course Description. This course will be a mathematical and philosophical exploration of infinity, covering a wide selection of topics illustrating this rich, fascinating concept—the mathematics and philosophy of the infinite.

Along the way, we shall find paradox and fun—and all my favorite elementary logic conundrums and puzzles. It will be part of my intention to reveal what I can of the quirky side of mathematics and logic in its connection with infinity, but with a keen eye open for when issues happen to engage with philosophically deeper foundational matters.

The lectures will be based on the chapters of my forthcoming book, The Book of Infinity, currently in preparation, and currently being serialized and made available on the Substack website as I explain below.

Topics. Among the topics we shall aim to discuss will be:

  • The Book of Numbers
  • Zeno’s paradox
  • The infinite coastline paradox
  • Supertasks
  • Largest number contest
  • The googol plex chitty bang stack hierarchy
  • Galileo’s Salviati on infinity
  • Hilbert’s Grand Hotel
  • The uncountable
  • How to count (to infinity and beyond!)
  • Slaying the Hydra
  • Transfinite recursion
  • The continuum hypothesis
  • The axiom of choice
  • Orders of infinity
  • The lattice of subsets of ℕ
  • Potential versus actual infinity
  • Confounding puzzles of infinity
  • Infinite liars
  • Infinite utilitarianism
  • Infinite computation
  • Infinite games
  • Indescribable numbers
  • Extremely remote events of enormous consequence
  • The sand reckoner
  • Paradox in high dimension
  • The outer limits of reason
  • Puzzles of epistemic logic and the problem of common knowledge

Mathematical background. The course will at times involve topics and concepts of a fundamentally mathematical nature, but no particular mathematical background or training will be assumed. Nevertheless, it is expected that students be open to mathematical thinking and ideas, and furthermore it is a core aim of the course to help develop the student’s mastery over various mathematical concepts connected with infinity.  

Readings. The lectures will be based on readings from the topic list above that will be made available on my Substack web page, Infinitely More. Readings for the topic list above will be gradually released there during the semester. Each reading will consist of a chapter essay my book-in-progress, The Book of Infinity, which is being serialized on the Substack site specifically for this course. In some weeks, there will be supplemental readings from other sources.

Student access. I will issue subscription invitations to the Substack site for all registered ND students using their ND email, with free access to the site during the semester, so that students can freely access the readings.  Students are free to manage their subscriptions however they see fit. Please inform me of any access issues. There are some excellent free Substack apps available for Apple iOS and Android for reading Substack content on a phone or other device.

Discussion forum. Students are welcome to participate in the discussion forums provided with the readings to discuss the topics, the questions, to post answer ideas, or engage in the discussion there. I shall try to participate myself by posting comments or hints.

Homework essays. Students are expected to engage fully with every topic covered in the class. Every chapter concludes with several Questions for Further Thought, with which the students should engage. It will be expected that students complete approximately half of the Questions for Further thought. Each question that is answered should be answered essay-style with a mini-essay of about half a page or more.

Extended essays. A student may choose at any time to answer one of the Questions for Further Thought more fully with a more extended essay of two or three pages, and in this case, other questions on that particular topic need not be engaged. Every student should plan to exercise this option at least twice during the semester.

Final exam.  There will be a final exam consisting of questions similar to those in the Questions for Further Thought, covering every topic that was covered in the course. The final grade will be based on the final exam and on the submitted homework solutions.

Open Invitation. Students outside of Notre Dame are welcome to follow along with the Infinity course, readings, and online discussion. Simply subscribe at Infinitely More, keep up with the readings and participate in the discussions we shall be having in the forums there.

Strategic thinking in infinite games, CosmoCaixa Science Museum, Barcelona, March 2023

I am deeply honored to be invited by la Caixa Foundation to give a talk in “The Greats of Science” talk series, to be held 16 March 2023 at the CosmoCaixa Science Museum in Barcelona. This talk series aspires to host “prestigious figures who have contributed towards admirable milestones, studies or discoveries,” who will bring the science to a general audience, aiming to “give viewers the chance to explore the most relevant parts of contemporary sicence through the top scientists of the moment.” Previous speakers include Jane Goodall and nearly a dozen Nobel Prize winners since 2018.

I hope to rise to those high expectations!

My topic will be: Strategic thinking in infinite games.

Have you time for an infinite game? Many familiar finite games admit natural infinitary analogues, infinite games that may captivate and challenge us with intriguing patterns and sublime complexity. Shall we have a game of infinite chess? Or how about infinite draughts, infinite Hex, infinite Wordle, or infinite Sudoku? In the Chocolatier’s game, the Chocolatier serves up an infinite stream of delicious morsels, while the Glutton aims to eat every one. These games and others illustrate the often subtle strategic aspects of infinite games, and sometimes their downright logical peculiarity. Does every infinite game admit of a winning strategy? Must optimal play be in principle computable? Let us discover the fascinating nature of infinitary strategic thinking.

The theory builds upon the classical finitary result of Zermelo (1913), the fundamental theorem of finite games, which shows that in every finite two-player game of perfect information, one of the players must have a winning strategy or both players have draw-or-better strategies. This result extends to certain infinitary games by means of the ordinal game-value analysis, which assigns transfinite ordinal values $\alpha$ to positions in a game, generalizing the familiar mate-in-$n$ idea of chess to the infinite. Current work realizes high transfinite game values in infinite chess, infinite draughts (checkers), infinite Go, and many other infinite games. The highest-known game value arising in infinite chess is the infinite ordinal $\omega^4$, and every countable ordinal arises in infinite draughts, the optimal result. Games exhibiting high transfinite ordinal game values have a surreal absurd character of play. The winning player will definitely win in finitely many moves, but the doomed losing player controls the process with absurdly long deeply nested patterns of forcing moves that must be answered, as though counting down from the infinite game value—when 0 is reached, the game is over.

Infinite Games, Frivolities of the Gods, Logic at Large Lecture, May 2022

The Dutch Association for Logic and Philosophy of the Exact Sciences (VvL) has organized a major annual public online lecture series called LOGIC AT LARGE, where “well-known logicians give public audience talks to a wide audience,” and I am truly honored to have been invited to give this year’s lecture. This will be an online event, the second of the series, scheduled for May 31, 2022 (note change in date!), and further access details will be posted when they become available. Free registration can be made on the VvL Logic at Large web page.

Abstract. Many familiar finite games admit natural infinitary analogues, which often highlight intriguing issues in infinite game theory. Shall we have a game of infinite chess? Or how about infinite draughts, infinite Hex, infinite Go, infinite Wordle, or infinite Sudoku? Let me introduce these games and use them to illustrate various fascinating concepts in the theory of infinite games.

Come enjoy the lecture, and stay for the online socializing event afterwards. Hope to see you there!

Infinite Wordle and the Mastermind numbers

  • J. D. Hamkins, “Infinite Wordle and the mastermind numbers,” Mathematics arXiv, 2022.
    author = {Joel David Hamkins},
    title = {Infinite Wordle and the mastermind numbers},
    journal = {Mathematics arXiv},
    year = {2022},
    volume = {},
    number = {},
    pages = {},
    month = {},
    note = {Under review},
    abstract = {},
    keywords = {under-review},
    source = {},
    doi = {},
    eprint = {2203.06804},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    url = {},

Download article at arXiv:2203.06804

Abstract. I consider the natural infinitary variations of the games Wordle and Mastermind, as well as their game-theoretic variations Absurdle and Madstermind, considering these games with infinitely long words and infinite color sequences and allowing transfinite game play. For each game, a secret codeword is hidden, which the codebreaker attempts to discover by making a series of guesses and receiving feedback as to their accuracy. In Wordle with words of any size from a finite alphabet of $n$ letters, including infinite words or even uncountable words, the codebreaker can nevertheless always win in $n$ steps. Meanwhile, the mastermind number 𝕞, defined as the smallest winning set of guesses in infinite Mastermind for sequences of length $\omega$ over a countable set of colors without duplication, is uncountable, but the exact value turns out to be independent of ZFC, for it is provably equal to the eventually different number $\frak{d}({\neq^*})$, which is the same as the covering number of the meager ideal $\text{cov}(\mathcal{M})$. I thus place all the various mastermind numbers, defined for the natural variations of the game, into the hierarchy of cardinal characteristics of the continuum.

Infinite Wordle and the mastermind numbers, CUNY Logic Workshop, March 2022

This will be an in-person talk for the CUNY Logic Workshop at the Graduate Center of the City University of New York on 11 March 2022.

Abstract. I shall introduce and consider the natural infinitary variations of Wordle, Absurdle, and Mastermind. Infinite Wordle extends the familiar finite game to infinite words and transfinite play—the code-breaker aims to discover a hidden codeword selected from a dictionary $\Delta\subseteq\Sigma^\omega$ of infinite words over a countable alphabet $\Sigma$ by making a sequence of successive guesswords, receiving feedback after each guess concerning its accuracy. For any dictionary using the usual 26-letter alphabet, for example, the code-breaker can win in at most 26 guesses, and more generally in $n$ guesses for alphabets of finite size $n$. Meanwhile, for some dictionaries on an infinite alphabet, infinite play is required, but the code-breaker can always win by stage $\omega$ on a countable alphabet, for any fixed dictionary. Infinite Mastermind, in contrast, is a subtler game than Wordle because only the number and not the position of correct bits is given. When duplication of colors is allowed, nevertheless, the code-breaker can still always win by stage $\omega$, but in the no-duplication variation, no countable number of guesses (even transfinite) is sufficient for the code-breaker to win. I therefore introduce the mastermind number, denoted $\frak{mm}$, defined to be the size of the smallest winning no-duplication Mastermind guessing set, a new cardinal characteristic of the continuum, which I prove is bounded below by the additivity number $\text{add}(\mathcal{M})$ of the meager ideal and bounded above by the covering number $\text{cov}(\mathcal{M})$. In particular, the precise value of the mastermind number is independent of ZFC and can consistently be strictly between $\aleph_1$ and the continuum $2^{\aleph_0}$. In simplified Mastermind, where the feedback given at each stage includes only the numbers of correct and incorrect bits (omitting information about rearrangements), then the corresponding simplified mastermind number is exactly the eventually different number $\frak{d}(\neq^*)$.

I am preparing an article on the topic, which will be available soon.

Infinite Hex is a draw

  • J. D. Hamkins and D. Leonessi, “Infinite Hex is a draw,” Mathematics arXiv, 2022.
    author = {Joel David Hamkins and Davide Leonessi},
    title = {Infinite Hex is a draw},
    journal = {Mathematics arXiv},
    year = {2022},
    volume = {},
    number = {},
    pages = {},
    month = {},
    note = {Under review},
    abstract = {},
    keywords = {under-review},
    source = {},
    doi = {},
    eprint = {2201.06475},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    url = {},

Download the article at

Abstract. We introduce the game of infinite Hex, extending the familiar finite game to natural play on the infinite hexagonal lattice. Whereas the finite game is a win for the first player, we prove in contrast that infinite Hex is a draw—both players have drawing strategies. Meanwhile, the transfinite game-value phenomenon, now abundantly exhibited in infinite chess and infinite draughts, regrettably does not arise in infinite Hex; only finite game values occur. Indeed, every game-valued position in infinite Hex is intrinsically local, meaning that winning play depends only on a fixed finite region of the board. This latter fact is proved under very general hypotheses, establishing the conclusion for all simple stone-placing games.

This is my second joint project with Davide Leonessi, the first being our work on Transfinite games values in infinite draughts, both projects growing out of his work on his MSc in MFoCS at Oxford, for which he earned a distinction in September 2021.

Here is a convenient online Hex player, for those who want to improve their game:

Infinite draughts and the logic of infinitary games, Oslo, November 2021

This will be a talk 11 November 2021 for the Oslo Seminar in Mathematical Logic, meeting online via Zoom at 10:15am CET (9:15am GMT) at Zoom: 671 7500 0197

Abstract. I shall give an introduction to the logic of infinite games, including the theory of transfinite game values, using the case of infinite draughts as a principal illustrative instance. Infinite draughts, also known as infinite checkers, is played like the finite game, but on an infinite checkerboard stretching without end in all four directions. In recent joint work with Davide Leonessi, we proved that every countable ordinal arises as the game value of a position in infinite draughts. Thus, there are positions from which Red has a winning strategy enabling her to win always in finitely many moves, but the length of play can be completely controlled by Black in a manner as though counting down from a given countable ordinal. This result is optimal for games having countably many options at each move—in short, the omega one of infinite draughts is true omega one.

Transfinite game values in infinite draughts

A joint paper with Davide Leonessi, in which we prove that every countable ordinal arises as the game value of a position in infinite draughts, and this result is optimal for games having countably many options at each move. In short, the omega one of infinite draughts is true omega one.

  • J. D. Hamkins and D. Leonessi, “Transfinite game values in infinite draughts,” Mathematics arXiv, 2021.
    author = {Joel David Hamkins and Davide Leonessi},
    title = {Transfinite game values in infinite draughts},
    journal = {Mathematics arXiv},
    year = {2021},
    volume = {},
    number = {},
    pages = {},
    month = {},
    note = {Under review},
    abstract = {},
    keywords = {under-review},
    source = {},
    doi = {},
    eprint = {2111.02053},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    url = {},

Download the paper at arXiv:2111.02053

Abstract. Infinite draughts, or checkers, is played just like the finite game, but on an infinite checkerboard extending without bound in all four directions. We prove that every countable ordinal arises as the game value of a position in infinite draughts. Thus, there are positions from which Red has a winning strategy enabling her to win always in finitely many moves, but the length of play can be completely controlled by Black in a manner as though counting down from a given countable ordinal.

Davide Leonessi, MSc MFoCS, Oxford, September 2021

Mr. Davide Leonessi successfully defended his dissertation for the Masters of Science degree in Mathematics and Foundations of Computer Science, entitled “Transfinite game values in infinite games,” on 15 September 2021. Davide earned a distinction for his thesis, an outstanding result.

Davide Leonessi | Google scholar | Dissertation | arXiv

Abstract. The object of this study are countably infinite games with perfect information that allow players to choose among arbitrarily many moves in a turn; in particular, we focus on the generalisations of the finite board games of Hex and Draughts.

In Chapter 1 we develop the theory of transfinite ordinal game values for open infinite games following [Evans-Hamkins 2014], and we focus on the properties of the omega one, that is the supremum of the possible game values, of classes of open games; we moreover design the class of climbing-through-$T$ games as a tool to study the omega one of given game classes.

The original contributions of this research are presented in the following two chapters.

In Chapter 2 we prove classical results about finite Hex and present Infinite Hex, a well-defined infinite generalisation of Hex.

We then introduce the class of stone-placing games, which captures the key features of Infinite Hex and further generalises the class of positional games already studied in the literature within the finite setting of Combinatorial Game Theory.

The main result of this research is the characterization of open stone-placing games in terms of the property of essential locality, which leads to the conclusion that the omega one of any class of open stone-placing games is at most $\omega$. In particular, we obtain that the class of open games of Infinite Hex has the smallest infinite omega one, that is $\omega_1^{\rm Hex}=\omega$.

In Chapter 3 we show a dual result; we define the class of games of Infinite Draughts and explicitly construct open games of arbitrarily high game value with the tools of Chapter 1, concluding that the omega one of the class of open games of Infinite Draughts is as high as possible, that is $\omega_1^{\rm Draughts}=\omega_1$.

The full dissertation is available:

My favorite theorem

What a pleasure it was to be interviewed by Evelyn Lamb and Kevin Knudson for their wonderful podcast series, My Favorite Theorem, available on Apple, Spotify, and any number of other aggregators.

I had a chance to talk about one my most favorite theorems, the fundamental theorem of finite games.

Theorem.(Zermelo 1913) In any two-player finite game of perfect information, one of the players has a winning strategy, or both players have drawing strategies.

Listen to the podcast here: My Favorite Theorem. A transcript is also available.

Infinite sets and Foundations—Interviewed on the Daniel Rubin Show

I was interviewed 26 August 2021 by mathematician Daniel Rubin on his show, and we had a lively, wideranging discussion spanning mathematics, infinity, and the philosophy of mathematics. Please enjoy!


0:00 Intro

2:11 Joel’s background. Interaction between math and philosophy

9:04 Joel’s work; infinite chess.

14:45 Infinite ordinals

22:27 The Cantor-Bendixson process

29:41 Uncountable ordinals

32:10 First order vs. second order theories

41:16 Non-standard analysis

46:57 The ZFC axioms and well-ordering of the reals

58:11 Showing independence of statements. Models and forcing.

1:04:38 Sets, classes, and categories

1:19:22 Is there one true set theory? Are projective sets Lebesgue measurable?

1:30:20 What does set theory look like if certain axioms are rejected?

1:36:06 How to judge philosophical positions about math

1:42:01 Concrete math where set theory becomes relevant. Tarski-Seidenberg on positive polynomials.

1:48:48 Goodstein sequences and the use of infinite ordinals

1:58:43 The state of set theory today

2:01:41 Joel’s recent books

Go check out the other episodes on Daniel’s channel!

Determinacy for proper class games, Seminaire de Logique Lyon-Paris, April 2021

This will be a talk for the Seminaire de Logique Lyon-Paris on 14 April 2021 4pm Paris time (3pm UK). The talk will be held on Zoom at
875 1148 7359

Abstract. The principle of open determinacy for class games — two-player games of perfect information with plays of length ω, where the moves are chosen from a possibly proper class, such as games on the ordinals — is not provable in Zermelo-Fraenkel set theory ZFC or Gödel-Bernays set theory GBC, if these theories are consistent, because provably in ZFC there is a definable open proper class game with no definable winning strategy. In fact, the principle of open determinacy and even merely clopen determinacy for class games implies Con(ZFC) and iterated instances Con(Con(ZFC)) and more, because it implies that there is a satisfaction class for first-order truth, and indeed a transfinite tower of truth predicates $\text{Tr}_\alpha$ for iterated truth-about-truth, relative to any class parameter. This is perhaps explained, in light of the Tarskian recursive definition of truth, by the more general fact that the principle of clopen determinacy is exactly equivalent over GBC to the principle of elementary transfinite recursion ETR over well-founded class relations. Meanwhile, the principle of open determinacy for class games is strictly stronger, although it is provable in the stronger theory GBC+$\Pi^1_1$-comprehension, a proper fragment of Kelley-Morse set theory KM.

Cantor’s Ice Cream Shoppe

Welcome to Cantor’s Ice Cream Shoppe! A huge choice of flavors—pile your cone high with as many scoops as you want!

Have two scoops, or three, four, or more! Why not infinitely many? Would you like $\omega$ many scoops, or $\omega\cdot2+5$ many scoops? You can have any countable ordinal number of scoops on your cone.

And furthermore, after ordering your scoops, you can order more scoops to be placed on top—all I ask is that you let me know how many such extra orders you plan to make. Let’s simply proceed transfinitely. You can announce any countable ordinal $\eta$, which will be the number of successive orders you will make; each order is a countable ordinal number of ice cream scoops to be placed on top of whatever cone is being assembled.

In fact, I’ll even let you change your mind about $\eta$ as we proceed, so as to give you more orders to make a taller cone.

So the process is:

  • You pick a countable ordinal $\eta$, which is the number of orders you will make.
  • For each order, you can pick any countable ordinal number of scoops to be added to the top of your ice-cream cone.
  • After making your order, you can freely increase $\eta$ to any larger countable ordinal, giving you the chance to make as many additional orders as you like.

At each limit stage of the ordering process, the ice cream cone you are assembling has all the scoops you’ve ordered so far, and we set the current $\eta$ value to the supremum of the values you had chosen so far.

If at any stage, you’ve used up your $\eta$ many orders, then the process has completed, and I serve you your ice cream cone. Enjoy!

Question. Can you arrange to achieve uncountably many scoops on your cone?

Although at each stage we place only countably many ice cream scoops onto the cone, nevertheless we can keep giving ourselves extra stages, as many as we want, simply by increasing $\eta$. Can you describe a systematic process of increasing the number of steps that will enable you to make uncountably many orders? This would achieve an unountable ice cream cone.

What is your solution? Give it some thought before proceeding. My solution appears below.

Alas, I claim that at Cantor’s Ice Cream Shoppe you cannot make an ice cream cone with uncountably many scoops. Specifically, I claim that there will inevitably come a countable ordinal stage at which you have used up all your orders.

Suppose that you begin by ordering $\beta_0$ many scoops, and setting a large value $\eta_0$ for the number of orders you will make. You subsequently order $\beta_1$ many additional scoops, and then $\beta_2$ many on top of that, and so on. At each stage, you may also have increased the value of $\eta_0$ to $\eta_1$ and then $\eta_2$ and so on. Probably all of these are enormous countable ordinals, making a huge ice cream cone.

At each stage $\alpha$, provided $\alpha<\eta_\alpha$, then you can make an order of $\beta_\alpha$ many scoops on top of your cone, and increase $\eta_\alpha$ to $\eta_{\alpha+1}$, if desired, or keep it the same.

At a limit stage $\lambda$, your cone has $\sum_{\alpha<\lambda}\beta_\alpha$ many scoops, and we update the $\eta$ value to the supremum of your earlier declarations $\eta_\lambda=\sup_{\alpha<\lambda}\eta_\alpha$.

What I claim now is that there will inevitably come a countable stage $\lambda$ for which $\lambda=\eta_\lambda$, meaning that you have used up all your orders with no possibility to further increase $\eta$. To see this, consider the sequence $$\eta_0\leq \eta_{\eta_0}\leq \eta_{\eta_{\eta_0}}\leq\cdots$$ We can define the sequence recursively by $\lambda_0=\eta_0$ and $\lambda_{n+1}=\eta_{\lambda_n}$. Let $\lambda=\sup_{n<\omega}\lambda_n$, the limit of this sequence. This is a countable supremum of countable ordinals and hence countable. But notice that $$\eta_\lambda=\sup_{n<\omega}\eta_{\lambda_n}=\sup_{n<\omega}\lambda_{n+1}=\lambda.$$ That is, $\eta_\lambda=\lambda$ itself, and so your orders have run out at $\lambda$, with no possibility to add more scoops or to increase $\eta$. So your order process completed at a countable stage, and you have therefore altogether only a countable ordinal number of scoops of ice cream. I’m truly very sorry at your pitiable impoverishment.

The theory of infinite games, including infinite chess, Talk Math With Your Friends, June 2020

This will be accessible online talk about infinite chess and other infinite games for the Talk Math With Your Friends seminar, June 18, 2020 4 pm EST (9 pm UK).  Zoom access information.  Please come talk math with me!

Abstract. I will give an introduction to the theory of infinite games, with examples drawn from infinite chess in order to illustrate various concepts, such as the transfinite game value of a position.

See more of my posts on infinite chess.