This will be a talk for the new Mathematical Philosophy Seminar here at the University of Notre Dame. Monday, September 21, 2026, 4pm, in Malloy Hall.
Title: How the continuum hypothesis could have been a fundamental axiom
Speaker: Joel David Hamkins, O’Hara Professor of Logic, Notre Dame
Abstract. I shall describe a simple historical thought experiment showing how our attitude toward the continuum hypothesis might easily have been very different than it is. If our mathematical history had been just a little different, I claim, if certain mathematical discoveries had been made in a slightly different order, then we would naturally have come to view the continuum hypothesis as a fundamental axiom of set theory, necessary for mathematics, indispensable even for the core ideas of calculus.
This will be a talk for the Logic Seminar at the University of Notre Dame on 15 September 2026, 2pm in the Hayes-Healy building room 125.
The contingent HOD dichotomy
Joel David Hamkins
Abstract: We shall discuss the contingently contingent nature of the class HOD of hereditarily ordinal-definable sets. In some models of set theory the axiom V=HOD is contingent by set forcing and in others it is not. After discussing some philosophical and historical puzzles concerning the nature of ordinal definability, I shall introduce and investigate what we call the contingent HOD dichotomy. Namely, it is provable in ZFC that either (1) the set-theoretic universe V is close to HOD in several respects: HOD is a ground model of V; the axiom V=HOD is forceable by set forcing; every object is ordinal-definable with a single additional parameter; and furthermore all these things are forcing invariant and hold throughout the generic multiverse; or (2) the universe is far from HOD; in particular V ≠ HOD; more generally, HOD is not a ground; the axiom V=HOD is not forceable; the universe is not ordinal-definable from a parameter; and these things hold invariantly throughout the generic multiverse. We shall explore how the dichotomy engages with large cardinals, with set-theoretic geology, with the iterated HOD^alpha models, with the maximality principles, and with Woodin’s HOD dichotomy.
This is new joint work in progress with Bokai Yao (Peking University).
This will be a talk for the CUNY Logic Workshop, 23 October 2026, 2pm, at the CUNY Graduate Center in midtown Manhattan.
The Contingent HOD Dichotomy
Joel David Hamkins, O’Hara Professor of Logic, University of Notre Dame
Abstract: We shall discuss the contingently contingent nature of the class HOD of hereditarily ordinal-definable sets. In some models of set theory the axiom V=HOD is contingent by set forcing and in others it is not. After discussing some philosophical and historical puzzles concerning the nature of ordinal definability, I shall introduce and investigate what we call the contingent HOD dichotomy. Namely, it is provable in ZFC that either (1) the set-theoretic universe V is close to HOD in several respects: HOD is a ground model of V; the axiom V=HOD is forceable by set forcing; every object is ordinal-definable with a single additional parameter; and furthermore all these things are forcing invariant and hold throughout the generic multiverse; or (2) the universe is far from HOD; in particular V ≠ HOD; more generally, HOD is not a ground; the axiom V=HOD is not forceable; the universe is not ordinal-definable from a parameter; and these things hold invariantly throughout the generic multiverse. We shall explore how the dichotomy engages with large cardinals, with set-theoretic geology, with the maximality principles, and with Woodin’s HOD dichotomy.
This is new joint work in progress with Bokai Yao (Peking University).
I am very pleased to announce that I have been awarded the Alternative Fields Medal in recognition of “Excellence in exposition of mathematics to a popular audience.” The award cites my various books as well as my participation on MathOverflow, mentioning that I have made nearly 2000 posts there since 2010, reaching 5.8 million viewers.
This will be a talk for the Fudan University logic group on 16 July 4:00pm in Shanghai.
Abstract: The axiom V = HOD is contingently contingent—more precisely, it is class-forcing contingent that V = HOD is contingent with respect to set forcing. It follows that V = HOD and □(V ≠ HOD) are logically independent of the maximality principles MP and MP(ℝ) over ZFC. I will give a full review of the relevant notions. This will be a chalkboard talk concerning very new joint work in progress with Bokai Yao (Peking University).
The paper is in progress and will be available I expect in a few weeks.
This is a talk for the Workshop on Mereology at Shandong University in Jinan, China, a part of the week-long conference Week of Fusion Philosophy, 22-26 June 2026. The mereology talks are on 22 June 2026.
Title: Set-theoretic mereology as a foundation of mathematics?
Speaker: Joel David Hamkins, University of Notre Dame, Peking University
Abstract. Mereology, the study of the relation of part to whole, is often contrasted with set theory and its membership relation, the relation of element to set. Whereas set theory has found success in the foundation of mathematics, since the time of Cantor, Zermelo and Hilbert, nevertheless mereology has been strangely absent. Why is this? In this talk, I shall introduce and discuss set-theoretic mereology, a form of mereology based upon the set-theoretic inclusion relation ⊆ rather than the element-of relation ∈. In particular, we shall investigate the role to be played by set-theoretic mereology in the foundations of mathematics, and come perhaps to an explanation of why it has been absent.
This will be a series of graduate lectures at Peking University, two lectures per week beginning mid-June and proceeding into July.
Topics. We shall aim to cover the central results in the modal logic of forcing, including an exploration of the main tools, and then explore how those ideas apply more generally in set-theoretic potentialism and other potentialist contexts, such as arithmetic potentialism and modal model theory. For prerequisites, students should be already familiar with some graduate-level set theory, including the basics of forcing, as well as standard tools from mathematical logic and model theory.
The lectures will focus on various research papers, as follows:
Joel David Hamkins. “A simple maximality principle.” Journal of Symbolic Logic 68.2 (2003),pp. 527-550. doi:10.2178/jsl/1052669062. arXiv:math/0009240.
Joel David Hamkins and Øystein Linnebo. “The Modal Logic of Set-theoretic Potentialism and the Potentialist Maximality Principles.” Review of Symbolic Logic 15.1 (2022), pp. 1-35. doi:10.1017/S1755020318000242. arXiv:1708.01644.
Joel David Hamkins and Wojciech Aleksander Wołoszyn. “Modal Model Theory.” Notre Dame Journal of Formal Logic 65.1 (2024), pp. 1-37. doi:10.1215/00294527-2024-0001.
Joel David Hamkins and Øystein Linnebo. “Second-order Potentialism.” research manuscript in preparatoin.
Joel David Hamkins. “The Modal Logic of Arithmetic Potentialism and the Universal Algorithm.” Philosophia Mathematica 34.1 (2026), pp. 137-182. doi:0.1093/philmat/nkag001.
Joel David Hamkins. “Every countable model of arithmetic or set theory has a pointwise definable end extension.” Kriterion Journal of Philosophy (2024). doi:10.1515/krt-2023-0029. arXiv:2209.12578.
Additional readings may be added, if time permits.
This will be a talk for the Philosophy Department Colloquium at Ohio University in Athens, OH on April 30th, 2026. I am very grateful for the invitation.
Abstract. Ultrafinitism is the philosophical view that only comparatively small or accessible numbers exist. I shall give an account of the deep model-theoretic connections between two otherwise very different-seeming approaches to ultrafinitism, which differ on the question of whether the feasible numbers are closed under successor. These connections are revealed and strengthened by adopting a potentialist outlook on the nature of arithmetic, where one’s realm of feasibility can be successively enlarged and enlarged again. This approach opens the door to a modal perspective on arithmetic and the idea of expressing core ultrafinitist principles in a modal vocabulary. Ultimately, this is an actualist modal model theory of ultrafinitist potentialism, which I take to shed light on the nature of ultrafinitism.
This will be a talk for the Logic Seminar at the University of Notre Dame, 14 April 2026, 2pm, Room 125 Hayes-Healey.
Abstract After establishing several general features of the hierarchy of consistency strength, we shall consider the possible spectrum of assertions of the form $n\in W$, where $W$ is a given computably enumerable set. If $W$ is c.e. but not computably decidable, many of these statements must be independent of PA, as well as ZFC, and indeed any consistent c.e. theory extending these. What kind of consistency strengths can be exhibited by these statements? In this work, we investigate the possible hierarchies of consistency strengths that arise. For example, there is a c.e. set $Q$ for which the consistency strengths of the assertions $n\in Q$ are linearly ordered like the rational line. More generally, I shall prove that every computable preorder relation on the natural numbers is realized exactly as the hierarchy of consistency strength for the membership statements $n\in W$ of some computably enumerable set $W$. After this, we shall consider the c.e. preorder relations. This is in part joint work with Atticus Stonestrom (Notre Dame).
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:
The Book of Numbers
The Sand Reckoner
Zeno’s Paradox
The Method of Exhaustion
Supertasks
The Infinite Coastline Paradox
The Paradox of Giants
The Paradox of the Largest Tweetable Number
Potential Versus Actual Infinity
Equinumerosity and Comparison of Size
What Is the Infinite?
Hilbert’s Grand Hotel
Uncountable Infinity
How to Count
Transfinite Recursive Constructions
Slaying the Hydra
The Continuum Hypothesis
Throwing Darts at the Real Line
The Orders of Infinity
The Surreal Numbers
The Axiom of Choice
Infinitary Hat Puzzles and the Aftermath
The Guessing-Box Puzzle
We Can Predict the Future
Infinite Liars
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.
Order now!
Meanwhile, I am serializing all my other books-in-progress on Infinitely More—subscribe now for full access to all my current work, including the surreal numbers, games, logic, philosophy of mathematics, and more.
I am honored to be invited to give the Owen G. Owens Memorial Lecture at Wayne State University on 16 April 2026, joining a distinguished list of luminaries giving previous Owens lectures, including Gregory Margulis, John Milnor, Mikhael Gromov, John Conway, and many others.
Abstract. Let us explore the theory of finite and infinite games, from the hypergame paradox to the fundamental theorem of finite games, which generalizes to vast classes of infinite games. We shall see how the ideas play out in infinite chess, infinite draughts (checkers), infinite Hex, infinite Wordle, and many other games.
I have been asked by the ASL to fill in as a last-minute substitute speaker for the ASL session at the upcoming 2026 APA Central Division Meeting in Chicago, February 18-21, 2026, due to a late cancellation of one of the plenary speakers, James Walsh, who regrettably is unable to speak. My talk will be part of the Wednesday evening ASL session 6-7:50.
Please join me in Chicago at the elegant Palmer House hotel—we have a great lineup of talks.
Title: Mathematicians do not agree on the essential structure of the complex numbers
Abstract: What is the essential structure of the complex numbers? Mathematicians, it turns out, do not generally agree—indeed one can find sharply worded disagreements. Do we have a purely algebraic conception of the complex numbers, taking it as an algebraically closed field with only its algebraic structure? Or do we have an analytic view, as a field over the real numbers, distinguished as a particular subfield? Or should we have a topological view? Perhaps we have a rigid conception of the complex plane, with the coordinate structure of real and imaginary parts. Many mathematicians find it fundamentally wrong to break the symmetry between i and -i, and indeed the various perspectives give rise to fundamentally different understandings of the automorphism group, and they are not all fully bi-interpretable nor even mutually interpretable. I shall place the whole discussion into the context of the philosophy of structuralism and the question of what is a number.
This will be a talk at the CUNY Logic Workshop on 13 March 2026, held at the CUNY Graduate Center.
Abstract. I shall introduce the elementary theory of surreal arithmetic (SA), a first-order theory that is true in the surreal field when equipped with its birthday order structure. This structure, I shall prove, is bi-interpretable with the set-theoretic universe (V,∈), and indeed the theory of surreal arithmetic SA is bi-interpretable with ZFC. This is joint work in progress with myself, Junhong Chen, and Ruizhi Yang, of Fudan University, Shanghai.
This will be a talk for the Logic Seminar at the University of Notre Dame, Tuesday 18 November 20215 2pm 125 Hayes-Healy Building.
Abstract. I shall introduce what I call the first-order elementary theory of surreal arithmetic, a theory that is true in the surreal field when equipped with its birthday order structure. This structure, I shall prove, is bi-interpretable with the set-theoretic universe (V,∈), and indeed the theory of surreal arithmetic SA is bi-interpretable with ZFC. This is very new joint work in progress with myself, Junhong Chen, and Ruizhi Yang, of Fudan University, Shanghai.
I am honored to be giving the 2025-26 Charles R. DePrima Memorial Lecture for the Mathematics Department of the California Institute of Technology. This lecture series aims to bring mathematical researchers to Caltech to give talks for a primarily undergraduate audience.
This invitation truly gives me a lot of pleasure, first, because Caltech is my alma mater (B.S. Mathematics 1988), but second, because my daughter is currently an undergraduate student at Caltech, majoring in mathematics. So I am looking forward to this talk.
