Abstract. What is or should be the potentialist account of classes? It turns out that there are several natural implementations of second-order logic in a modal potentialist setting, which arise from differing philosophical conceptions of the nature of the second-order resources. I shall introduce the proposals, analyze their comparative expressive and interpretative powers, and explain how various philosophical attitudes are fulfilled or not for each proposal. This is joint work in progress with Øystein Linnebo.

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 dleonessi@gc.cuny.edu 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.

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.

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

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

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

We had a sweeping discussion touching upon many issues in the philosophy of mathematics, including the nature of mathematical truth, mathematical abstraction, the nature of mathematical existence, the meaning and role of proof in mathematics, the completeness theorem, the incompleteness phenomenon, infinity, and a discussion about the motivations that one might have for studying mathematics.

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.

This will be a talk for the Axe Histoire et Philosophie des mathématiques, Séminaire PhilMath Intersem 2023, a collaborative event sponsored by the University of Notre Dame and le laboratoire SPHERE, Paris. The Intersem runs several weeks, but my talk will be 9 June.

Abstract. The 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 εF, in such a way that Basic Law V is fulfilled:

εF=εG ⇔ ∀x (Fx ⇔ 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, one amounting, I argue, to a truth predicate in Frege’s system. The main result therefore leads to a proof of Tarski’s theorem on the nondefinability of truth as a corollary to Russell’s argument, independently of Gödel. 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.

Abstract. I shall speak on the surprising strength of the second-order reflection principle in the context of set theory with abundant urelements. The theory GBcU with the abundant urelement axiom and second-order reflection is bi-interpretable with a strengthening of KM with a supercompact cardinal. This is joint work with Bokai Yao.

Abstract. There is an unexplained logical mystery in the foundations of mathematics, namely, our best and strongest mathematical theories seem to be linearly ordered and indeed well-ordered by consistency strength. Why should it be? The phenomenon is thought to carry significance for foundations, perhaps pointing us, some have argued, toward the ultimately correct mathematical theories, the “one road upward.” And yet, we know as a purely formal matter that the hierarchy of consistency strength is not well-ordered. It is ill-founded, densely ordered, and nonlinear. The statements commonly used to illustrate these features, however, are often dismissed as unnatural or as Gödelian trickery. In this talk, however, I aim to rebut that criticism by presenting a variety of natural hypotheses that reveal illfoundedness in consistency strength, density in the hierarchy of consistency strength, and incomparability in consistency strength. This will lead to discussion of the role and meaning of “natural” in the foundations of mathematics.

The meeting will be in person and online. Those who wish to attend via Zoom, please write to Daniel Isaacson.

This will be a talk (in person) for the Logic Seminar of the Mathematics Institute of the Univerisity of Oxford, May 18, 2023 5pm, Wiles Building L3.

Abstract: I shall present a new flexible method showing that every countable model of PA admits a pointwise definable end-extension, one in which every point is definable without parameters. Also, any model of PA of size at most continuum admits an extension that is Leibnizian, meaning that any two distinct points are separated by some expressible property. Similar results hold in set theory, where one can also achieve V=L in the extension, or indeed any suitable theory holding in an inner model of the original model.

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.

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.

Abstract: I shall explore several senses in which set-theoretic forcing can be seen as a computational process on the models of set theory. Given an oracle for the atomic or elementary diagram of a model (M,∈M) of set theory, for example, there are senses in which one may compute M-generic filters G⊂ℙ∈M over that model and compute the diagrams of the corresponding forcing extensions M[G]. Meanwhile, no such computational process is functorial, for there must always be isomorphic alternative presentations of the same model of set theory that lead by the computational process to non-isomorphic forcing extensions. Indeed, there is no Borel function providing generic filters that is functorial in this sense. This is joint work with myself, Russell Miller and Kameryn Williams.

The talk took place in “The Barn” in the upper space between the Reyerson Laboratory and Eckhart Hall, where the University of Chicago Department of Mathematics is located:

Abstract: I shall survey the surprisingly enormous variety of potentialist conceptions, even in the case of arithmetic potentialism, spanning a spectrum from linear inevitabilism and other convergent potentialist conceptions to more radical nonamalgamable branching-possibility potentialist conceptions. Underlying the universe-fragment framework for potentialism, one finds a natural modal vocabulary capable of expressing fine distinctions between the various potentialist ideas, as well as sweeping potentialist principles. Similarly diverse conceptions of ultrafinitism grow out of the analysis. Ultimately, the various convergent potentialist conceptions, I shall argue, are implicitly actualist, reducing to and interpreting actualism via the potentialist translation, whereas the radical-branching nonamalgamable potentialist conception admits no such reduction.

I am a commentator at the Pacific APA 2023 conference in San Francisco in a Book Symposium session focused on the book of Catarina Duthil Novaes, The Dialogical Roots of Deduction.

I think very highly of Novaes’s book (my book review is here) and I nominated it for the Lakatos prize, which I am very glad to say that she won. This is particularly appropriate in my view in light of Lakatos’s own use of dialogues in expressing his perspectives on the philosophy of mathematics and the nature of proof.

I shall be a speaker at the book symposium, intending to place the dialogical perspective on proof in the context of a variety of other views of proof. I shall conclude with a few criticisms of the book, which I hope might lead to interesting discussion.

Abstract. I shall present a new flexible method showing that every countable model of PA admits a pointwise definable-elementary end-extension. Also, any model of PA of size at most continuum admits an extension that is Leibnizian, meaning that any two distinct points are separated by some expressible property. Similar results hold in set theory, where one can also achieve V=L in the extension, or indeed any suitable theory holding in an inner model of the original model.

This will be a talk 15 March 2023 for the Mathematics Department of the University of Barcelona, organized jointly with the Set Theory Seminar.

Abstract. According to the math tea argument, perhaps heard at a good afternoon tea, there must be some real numbers that we can neither describe nor define, since there are uncountably many real numbers, but only countably many definitions. Is it correct? In this talk, I shall discuss the phenomenon of pointwise definable structures in mathematics, structures in which every object has a property that only it exhibits. A mathematical structure is Leibnizian, in contrast, if any pair of distinct objects in it exhibit different properties. Is there a Leibnizian structure with no definable elements? We shall discuss many interesting elementary examples, eventually working up to the proof that every countable model of set theory has a pointwise definable extension, in which every mathematical object is definable, including every real number, every function, every set. We shall discuss the relevance for the math tea argument.