Tag Archives: CUNY

Epistemic Logic Puzzles for Mathematicians, NYC Math Education and Communication Seminar, October 2026

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This will be a talk for the NYC Math Education and Communication Seminar, 20 October 2026, 12:15 – 1:15pm, at the CUNY Graduate Center in midtown Manhattan.

Epistemic Logic Puzzles for Mathematicians

Joel David Hamkins, O’Hara Professor of Logic, Notre Dame

Abstract: We are guests at the Midnight Ball, amongst infinitely many friends, and everyone has a natural number written on their forehead. We can see the other numbers, but not our own. The Queen of the Ball is chosen, and she alone is allowed to change her number to whatever she likes. At midnight, we are to shout our best guess for our own number. With advance planning— but no communication after we have our numbers—how well can we do? Find out at the talk, where we shall also explore many other puzzles. We shall meet the blue-eyed islanders, the pirates dividing their treasure, and the Machiavellian philosopher’s council. Some solutions will rely on the axiom of choice. I shall conclude with the nearly perfect prediction theorem, showing how to predict the current and future values of an unknown (possibly discontinuous) function on the real numbers, with almost-everywhere perfect accuracy, based solely on the history of prior values. After the talk in the Q&A, we shall discuss specific methods and techniques of mathematical exposition and outreach—I shall look forward to this very much.

Slides – Epistemic Logic Puzzles for Mathematicians – Hamkins – CUNY GC Oct 2026Download

On the fundamental theorem of finite games, CUNY Student Logic Seminar, October 2026

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This will be a talk for the CUNY Student Logic Seminar on 23rd October 2026, 5pm at the CUNY Graduate Center in midtown Manhattan.

On the fundamental theorem of finite games

Abstract: We will give five different proofs of the fundamental theorem of finite games, due originally to Zermelo (1913), showing that in any two-player game of perfect information, one of the players has a winning strategy or both players have drawing strategies. We will discuss how the proofs generalize to infinite games, giving several different proofs of the Gale-Stewart theorem on open determinacy, and also establish the existence of nondetermined infinite games. 

The talk will be based in part of my “Short Stories” essay that appeared in the AMS Notices:

  • Joel David Hamkins. The fundamental theorem of finite games. Notices AMS 72.8 (2025). Short Stories column, pp. 858-860. doi:10.1090/noti3203.

See also my substack essay:

In the talk, I will go beyond both these essays, to speak of game indeterminacy and the proof from the axiom of choice that there are nondetermined games.

The Contingent HOD Dichotomy, CUNY Logic Workshop, October 2026

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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).

Slides – The Contingent HOD Dichotomy – CUNY 2026Download

Surreal arithmetic is bi-interpretable with set theory, CUNY Logic Workshop, March 2026

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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.

Slides – Elementary theory of surreal arithmetic – Hamkins – CUNY March 2026Download

Realizing Frege’s Basic Law V provably in ZFC, New York, May 2023

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This will be a talk for the CUNY Set Theory Seminar on May 5, 2023 10am. Contact the organizers for the Zoom link.

Abstract. The standard set-theoretic distinction between sets and classes instantiates in important respects the Fregean distinction between objects and concepts, for in set theory we commonly take the universe of sets as a realm of objects to be considered under the guise of diverse concepts, the definable classes, each serving as a predicate on that domain of individuals. Although it is commonly held that in a very general manner, there can be no association of classes with objects in a way that fulfills Frege’s Basic Law V, nevertheless, in the ZF framework, it turns out that we can provide a completely deflationary account of this and other Fregean abstraction principles. Namely, there is a mapping of classes to objects, definable in set theory in senses I shall explain (hence deflationary), associating every first-order parametrically definable class $F$ with a set object $\varepsilon F$, in such a way that Basic Law V is fulfilled: 

 $$\varepsilon F =\varepsilon G\iff\forall x\ (Fx\leftrightarrow Gx).$$ 

Russell’s elementary refutation of the general comprehension axiom, therefore, is improperly described as a refutation of Basic Law V itself, but rather refutes Basic Law V only when augmented with powerful class comprehension principles going strictly beyond ZF. The main result leads also to a proof of Tarski’s theorem on the nondefinability of truth as a corollary to Russell’s argument. A central goal of the project is to highlight the issue of definability and deflationism for the extension assignment problem at the core of Fregean abstraction.