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.
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!
This will be a talk for the conference Fudan Model Theory and Philosophy of Mathematics, held at Fudan University in Shanghai and online, 21-24 August 2021. My talk will take place on Zoom on 23 August 20:00 Beijing time (1pm BST).
Abstract. Tennenbaum famously proved that there is no computable presentation of a nonstandard model of arithmetic or indeed of any model of set theory. In this talk, I shall discuss the Tennenbaum phenomenon as it arises for computable quotient presentations of models. Quotient presentations offer a philosophically attractive treatment of identity, a realm in which questions of identity are not necessarily computable. Objects in the presentation serve in effect as names for objects in the final quotient structure, names that may represent the same or different items in that structure, but one cannot necessarily tell which. Bakhadyr Khoussainov outlined a sweeping vision for quotient presentations in computable model theory and made several conjectures concerning the Tennenbaum phenomenon. In this talk, I shall discuss joint work with Michał Godziszewski that settles and addresses several of these conjectures.
This is a talk for the Logik Kolloquium at the University of Konstanz, spanning the departments of mathematics, philosophy, linguistics, and computer science. 19 July 2021 on Zoom. 15:15 CEST (2:15 pm BST).
Abstract: An enduring mystery in the foundations of mathematics is the observed phenomenon that our best and strongest mathematical theories seem to be linearly ordered and indeed well-ordered by consistency strength. For any two of the familiar large cardinal hypotheses, one of them generally proves the consistency of the other. Why should this be? Why should it be linear? Some philosophers see the phenomenon as significant for the philosophy of mathematics—it points us toward an ultimate mathematical truth. Meanwhile, the linearity phenomenon is not strictly true as mathematical fact, for we can prove that the hierarchy of consistency strength is actually ill-founded, densely ordered, and nonlinear. The counterexample statements and theories, however, are often dismissed as unnatural. Linearity is thus a phenomenon only for the so-called “naturally occurring” theories. But what counts as natural? Is there a mathematically meaningful account of naturality? In this talk, I shall criticize this notion of naturality, and attempt to undermine the linearity phenomenon by presenting a variety of natural hypotheses that reveal ill-foundedness, density, and incomparability in the hierarchy of consistency strength.
The talk should be generally accessible to university logic students.
Abstract. Zermelo famously proved that second-order ZFC is quasi-categorical—the models of this theory are precisely the rank-initial segments of the set-theoretic universe cut off at an inaccessible cardinal. Which are the fully categorical extensions of this theory? This question gives rise to the notion of categorical large cardinals, and opens the door to several puzzling philosophical issues, such as the conflict between categoricity as a fundamental value in mathematics and reflection principles in set theory. (This is joint work with Robin Solberg, Oxford.)
This will be a talk for the Department Colloquium of the Philosophy Department of the University of Notre Dame, 26 March 12 pm EST (4pm GMT).
Abstract: Potentialism is the view, originating in the classical dispute between actual and potential infinity, that one’s mathematical universe is never fully completed, but rather unfolds gradually as new parts of it increasingly come into existence or become accessible or known to us. Recent work emphasizes the modal aspect of potentialism, while decoupling it from arithmetic and from infinity: the essence of potentialism is about approximating a larger universe by means of universe fragments, an idea that applies to set-theoretic as well as arithmetic foundations. The modal language and perspective allows one precisely to distinguish various natural potentialist conceptions in the foundations of mathematics, whose exact modal validities are now known. Ultimately, this analysis suggests a refocusing of potentialism on the issue of convergent inevitability in comparison with radical branching. I shall defend the theses, first, that convergent potentialism is implicitly actualist, and second, that we should understand ultrafinitism in modal terms as a form of potentialism, one with surprising parallels to the case of arithmetic potentialism.
This will be an event for the $\Phi$-Math Reading Group at the Institute for Logic, Language, and Computation (ILLC) at the University of Amsterdam, 19 March 2021 6pm CET (5pm GMT). Zoom access here.
I shall make a brief presentation of the overall contents of the book, including a discussion of my perspective on the subject, and then get into some of the detailed issues with which the book engages. After this, we shall open up for discussion and comments.
This was a talk for the Kobe Set Theory Workshop, held on the occasion of Sakaé Fuchino’s retirement, 9-11 March 2021.
Abstract. I shall discuss 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 of set theory $\langle M,\in^M\rangle$, for example, one may in various senses compute $M$-generic filters $G\subset P\in M$ and 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 $M$ that lead by the computational process to non-isomorphic forcing extensions $M[G]\not\cong M[G’]$. Indeed, there is no Borel function providing generic filters that is functorial in this sense.
This is joint work with Russell Miller and Kameryn Williams.
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.
This will be a talk for a new mathematical logic seminar at the University of Warsaw in the Department of Hhilosophy, entitled Epistemic and Semantic Commitments of Foundational Theories, devoted to formal truth theories and implicit commitments of foundational theories as well as their conceptual surroundings.
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.
The talk will be held online via Zoom ID: 998 6013 7362.
Abstract. It is a mystery often mentioned in the foundations of mathematics that our best and strongest mathematical theories seem to be linearly ordered and indeed well-ordered by consistency strength. Given any two of the familiar large cardinal hypotheses, for example, generally one of them proves the consistency of the other. Why should this be? The phenomenon is seen as significant for the philosophy of mathematics, perhaps pointing us toward the ultimately correct mathematical theories. 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 usually used to illustrate these features are often dismissed as unnatural or as Gödelian trickery. In this talk, I aim to overcome that criticism—as well as I am able to—by presenting a variety of natural hypotheses that reveal ill-foundedness in consistency strength, density in the hierarchy of consistency strength, and incomparability in consistency strength.
The talk should be generally accessible to university logic students, requiring little beyond familiarity with the incompleteness theorem and some elementary ideas from computability theory.
Abstract. Recent years have seen a flurry of mathematical activity in set-theoretic and arithmetic potentialism, in which we investigate a collection of models under various natural extension concepts. These potentialist systems enable a modal perspective—a statement is possible in a model, if it is true in some extension, and necessary, if it is true in all extensions. We consider the models of ZFC set theory, for example, with respect to submodel extensions, rank-extensions, forcing extensions and others, and these various extension concepts exhibit different modal validities. In this talk, I shall describe the state of current developments, including the most recent tools and results.
This will be a talk for the Models of Peano Arithmetic (MOPA) seminar on 11 November 2020, 12 pm EST (5pm GMT). Kindly note the rescheduled date and time.
Abstract. Ali Enayat had asked whether there is a model of Peano arithmetic (PA) that can be represented as $\newcommand\Q{\mathbb{Q}}\langle\Q,\oplus,\otimes\rangle$, where $\oplus$ and $\otimes$ are continuous functions on the rationals $\Q$. We prove, affirmatively, that indeed every countable model of PA has such a continuous presentation on the rationals. More generally, we investigate the topological spaces that arise as such topological models of arithmetic. The reals $\mathbb{R}$, the reals in any finite dimension $\mathbb{R}^n$, the long line and the Cantor space do not, and neither does any Suslin line; many other spaces do; the status of the Baire space is open.
This is joint work with Ali Enayat, myself and Bartosz Wcisło.
This will be a talk for the Barcelona Set Theory Seminar, 28 October 2020 4 pm CET (3 pm UK). Contact Joan Bagaria bagaria@ub.edu for the access link.
Abstract. The Barwise extension theorem, asserting that every countable model of ZF set theory admits an end-extension to a model of ZFC+V=L, is both a technical culmination of the pioneering methods of Barwise in admissible set theory and infinitary logic and also one of those rare mathematical theorems that is saturated with philosophical significance. In this talk, I shall describe a new proof of the theorem that omits any need for infinitary logic and relies instead only on classical methods of descriptive set theory. This proof leads directly to the universal finite sequence, a Sigma_1 definable finite sequence, which can be extended arbitrarily as desired in suitable end-extensions of the universe. The result has strong consequences for the nature of set-theoretic potentialism. This work is joint with Kameryn J. Williams.
This will be a talk for the Oslo potentialism workshop, Varieties of Potentialism, to be held online via Zoom on 23 September 2020, from noon to 18:40 CEST (11am to 17:40 UK time). My talk is scheduled for 13:10 CEST (12:10 UK time). Further details about access and registration are availavle on the conference web page.
Abstract. I shall introduce and describe the subject of modal model theory, in which one studies a mathematical structure within a class of similar structures under an extension concept, giving rise to mathematically natural notions of possibility and necessity, a form of mathematical potentialism. We study the class of all graphs, or all groups, all fields, all orders, or what have you; a natural case is the class $\text{Mod}(T)$ of all models of a fixed first-order theory $T$. In this talk, I shall describe some of the resulting elementary theory, such as the fact that the $\mathcal{L}$ theory of a structure determines a robust fragment of its modal theory, but not all of it. The class of graphs illustrates the remarkable power of the modal vocabulary, for the modal language of graph theory can express connectedness, colorability, finiteness, countability, size continuum, size $\aleph_1$, $\aleph_2$, $\aleph_\omega$, $\beth_\omega$, first $\beth$-fixed point, first $\beth$-hyper-fixed-point and much more. When augmented with the actuality operator @, modal graph theory becomes fully bi-interpretable with truth in the set-theoretic universe. This is joint work with Wojciech Wołoszyn.
