Infinite-Games Workshop

Welcome to the Infinite-Games Workshop, beginning Autumn 2023. The past ten years has seen an explosion in the study of infinite games, for researchers are now investigating diverse infinite games, including infinite chess, infinite draughts, infinite Hex, infinite Othello, infinite Go, indeed, we seem to have research projects involving infinitary analogues of all our familiar finite games. It is an emerging research area with many new exciting results.

This autumn, we shall set the workshop off with talks on several exciting new results in infinite chess, results which settle what had been some of the big open questions in the topic, including the question of the omega one of chess—the supremum of the ordinal game values that arise—as well as a finite position with game value $\omega^2$.

The workshop talks will be run at a high level of sophistication, aimed for the most part at serious researchers currently working in this emerging area. Mathematicians, computer scientists, infinitary game theorists, all serious researchers are welcome.

All talks will take place on Zoom at meeting 968 0186 3645 (password = latex code for the first uncountable ordinal). Contact for further information.

Talks will be 90 minutes, with a workshop style welcoming questions. All talks will be recorded and placed on our YouTube channel. Talks will generally be held on Thursdays at 11:00 am New York time.

Add our calendar: Infinite-Games Workshop Calendar

The workshop is being organized by myself with the assistance of Davide Leonessi.

FAll 2023 Talks

21 September 2023 11am EST

Infinite draughts: a solved open game

Davide Leonessi, The Graduate Center of the City University of New York

Davide Leonessi, CUNY GC

Abstract: In this talk I will introduce open infinite games, and then define a natural generalization of draughts (checkers) to the infinite planar board. Infinite draughts is an open game, giving rise to the game value phenomenon and expressing it fully—the omega one of draughts is at least true $\omega_1$ and every possible defensive strategy of the losing player can be implemented. 

5 October 2023 11am EST

Introduction to infinite games

Joel David Hamkins, Professor of Logic, University of Notre Dame

Abstract: I shall give a general introduction to the subject and theory of infinite games, drawing upon diverse examples of infinitary games, but including also infinite chess, infinite Hex, infinite draughts, and others.

26 October 2023 11am EST

Complexity of the winning condition of infinite Hex

Ilkka Törmä, University of Turku, Finland

Abstract: Hex is a two-player game where the goal is to form a contiguous path of tokens from one side of a finite rectangular board to the opposite side. It is a famous classical result that Hex admits no draws: a completely filled board is a win for exactly one player. Infinite Hex is a variant introduced recently by Hamkins and Leonessi. It is played on the infinite two-dimensional grid $\mathbb{Z}^2$, and a player wins by forming a certain kind of two-way infinite contiguous path. Hamkins and Leonessi left open the complexity of the winning condition, in particular whether it is Borel. We present a proof that it is in fact arithmetic.

16 NOvember 2023 11am EST

A finite position in infinite chess with game value $\omega^2+k$

Andreas Tsevas, Physics, Ludwig Maximalians Universität München

Abstract: I present a position in infinite chess with finitely many pieces and a game value of $\omega^2+k$ for $k\in\mathbb N$, thereby improving the previously known best result in the finite case of $\omega\cdot n$ for arbitrary $n \in\mathbb N$. This is achieved by exercising control over the movement of a white queen along two rows on the chessboard via precise tempo manipulation and utilization of the uniquely crucial ability of the queen to interlace horizontal threats with diagonal moves.

7 December 2023 11am EST

All Countable Ordinals Arise as Game Values in Infinite Chess

Matthew Bolan, University of Toronto

Abstract: For every countable ordinal $\alpha$, we show that there exists a position in infinite chess with infinitely many pieces having game value $\alpha$.

An exploration of infinite games—infinite Wordle and the Mastermind numbers, Harvard, October 2023

This will be a talk 16 October 2023 (Note new date!) for the Colloquium of the Harvard Center for Mathematical Sciences and Applications (CMSA).

Abstract: Let us explore the nature of strategic reasoning in infinite games, focusing on the cases of infinite Wordle and infinite Mastermind. The familiar game of Wordle extends naturally to longer words or even infinite words in an idealized language, and Mastermind similarly has natural infinitary analogues. What is the nature of play in these infinite games? Can the codebreaker play so as to win always at a finite stage of play? The analysis emerges gradually, and in the talk I shall begin slowly with some easy elementary observations. By the end, however, we shall engage with sophisticated ideas in descriptive set theory, a kind of infinitary information theory. Some assertions about the minimal size of winning sets of guesses, for example, turn out to be independent of the Zermelo-Fraenkel ZFC axioms of set theory. Some questions remain open.

Infinite games—strategies, logic, theory, and computation, Northeastern, June 2023

This will be an online Zoom talk for the Boston Computaton Club, a graduate seminar in computer science at Northeastern University, 16 June 12pm EST (note change in date/time). Contact the organizers for the Zoom link.

Abstract: Many familiar finite games admit natural infinitary analogues, which may captivate and challenge us with 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.

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.

Cosmo Caixa announcement

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!