This will be accessible online talk about infinite chess and other infinite games for the Talk Math With Your Friends seminar, June 18, 2020 4 pm EST (9 pm UK). Zoom access information. Please come talk math with me!

Abstract. I will give an introduction to the theory of infinite games, with examples drawn from infinite chess in order to illustrate various concepts, such as the transfinite game value of a position.

This will be a talk for the newly founded Oxford Set Theory Seminar, May 20, 2020. Contact Sam Adam-Day (me@samadamday.com) for the Zoom access codes.

Abstract: Set theory exhibits a truly robust mutual interpretability phenomenon: in any model of one set theory we can define models of diverse other set theories and vice versa. In any model of ZFC, we can define models of ZFC + GCH and also of ZFC + ¬CH and so on in hundreds of cases. And yet, it turns out, in no instance do these mutual interpretations rise to the level of bi-interpretation. Ali Enayat proved that distinct theories extending ZF are never bi-interpretable, and models of ZF are bi-interpretable only when they are isomorphic. So there is no nontrivial bi-interpretation phenomenon in set theory at the level of ZF or above. Nevertheless, for natural weaker set theories, we prove, including ZFC- without power set and Zermelo set theory Z, there are nontrivial instances of bi-interpretation. Specifically, there are well-founded models of ZFC- that are bi-interpretable, but not isomorphic—even $\langle H_{\omega_1},\in\rangle$ and $\langle H_{\omega_2},\in\rangle$ can be bi-interpretable—and there are distinct bi-interpretable theories extending ZFC-. Similarly, using a construction of Mathias, we prove that every model of ZF is bi-interpretable with a model of Zermelo set theory in which the replacement axiom fails. This is joint work with Alfredo Roque Freire.

This is a version of the talk that I had planned to give at the 2020 Set Theory meeting Oberwolfach, before that meeting was canceled on account of the Covid-19 situation.

I was interviewed by Theodor Nenu as the first installment of his Philosophical Trials interview series with philosophers, mathematicians and physicists.

Theodor provided the following outline of the conversation:

Note: the conference has been cancelled due to concerns over the Coronavirus-19. (Meanwhile, I have given the talk for the Oxford Set Theory Seminar — see below.)

Abstract: Set theory exhibits a truly robust mutual interpretability phenomenon: in any model of one set theory we can define models of diverse other set theories and vice versa. In any model of ZFC, we can define models of ZFC + GCH and also of ZFC + ¬CH and so on in hundreds of cases. And yet, it turns out, in no instance do these mutual interpretations rise to the level of bi-interpretation. Ali Enayat proved that distinct theories extending ZF are never bi-interpretable, and models of ZF are bi-interpretable only when they are isomorphic. So there is no nontrivial bi-interpretation phenomenon in set theory at the level of ZF or above. Nevertheless, for natural weaker set theories, we prove, including ZFC- without power set and Zermelo set theory Z, there are nontrivial instances of bi-interpretation. Specifically, there are well-founded models of ZFC- that are bi-interpretable, but not isomorphic—even $\langle H_{\omega_1},\in\rangle$ and $\langle H_{\omega_2},\in\rangle$ can be bi-interpretable—and there are distinct bi-interpretable theories extending ZFC-. Similarly, using a construction of Mathias, we prove that every model of ZF is bi-interpretable with a model of Zermelo set theory in which the replacement axiom fails. This is joint work with Alfredo Roque Freire.

Abstract: In contrast to the robust mutual interpretability phenomenon in set theory, Ali Enayat proved that bi-interpretation is absent: distinct theories extending ZF are never bi-interpretable and models of ZF are bi-interpretable only when they are isomorphic. Nevertheless, for natural weaker set theories, we prove, including Zermelo-Fraenkel set theory ZFC- without power set and Zermelo set theory Z, there are nontrivial instances of bi-interpretation. Specifically, there are well-founded models of ZFC- that are bi-interpretable, but not isomorphic—even $\langle H_{\omega_1},\in\rangle$ and $\langle H_{\omega_2},\in\rangle$ can be bi-interpretable—and there are distinct bi-interpretable theories extending ZFC-. Similarly, using a construction of Mathias, we prove that every model of ZF is bi-interpretable with a model of Zermelo set theory in which the replacement axiom fails. This is joint work with Alfredo Roque Freire.

This will be a fun talk for the Philosophy Plus Science Taster Day, a fun day of events for prospective students in the joint philosophy degrees, whether Mathematics & Philosophy, Physics & Philosophy or Computer Science & Philosophy. The talk will be Friday 10th January in the Andrew Wiles building.

Abstract. In this talk, we shall pose and solve various fun puzzles in epistemic logic, which is to say, puzzles involving reasoning about knowledge, including one’s own knowledge or the knowledge of other people, including especially knowledge of knowledge or knowledge of the lack of knowledge. We’ll discuss several classic puzzles of common knowledge, such as the two-generals problem, Cheryl’s birthday problem, and the blue-eyed islanders, as well as several new puzzles. Please come and enjoy!

This will be my talk for the Set Theory in the United Kingdom 4, a conference to be held in Oxford on 14 December 2019. I am organizing the conference with Sam Adam-Day.

Modal model theory

Abstract. I shall introduce the subject of modal model theory, a research effort bringing modal concepts and vocabulary into model theory. For any first-order theory T, we may naturally consider the models of T as a Kripke model under the submodel relation, and thereby naturally expand the language of T to include the modal operators. In the class of all graphs, for example, a statement is possible in a graph, if it is true in some larger graph, having that graph as an induced subgraph, and a statement is necessary when it is true in all such larger graphs. The modal expansion of the language is quite powerful: in graphs it can express k-colorability and even finiteness and countability. The main idea applies to any collection of models with an extension concept. The principal questions are: what are the modal validities exhibited by the class of models or by individual models? For example, a countable graph validates S5 for graph theoretic assertions with parameters, for example, just in case it is the countable random graph; and without parameters, just in case it is universal for all finite graphs. Similar results apply with digraphs, groups, fields and orders. This is joint work with Wojciech Wołoszyn.

This will be a fun start-of-term Philosophy Undergraduate Welcome Lecture for philosophy students at Oxford in the Mathematics & Philosophy, Physics & Philosophy, Computer Science & Philosophy, and Philosophy & Linguistics degrees. New students are especially encouraged, but everyone is welcome! The talk is open to all. The talk will be Wednesday 16th October, 5-6 pm in the Mathematical Institute, with wine and nibbles afterwards.

Abstract. In this talk, we shall pose and solve various fun puzzles in epistemic logic, which is to say, puzzles involving reasoning about knowledge, including one’s own knowledge or the knowledge of other people, including especially knowledge of knowledge or knowledge of the lack of knowledge. We’ll discuss several classic puzzles of common knowledge, such as the two-generals problem, Cheryl’s birthday problem, and the blue-eyed islanders, as well as several new puzzles. Please come and enjoy!

I shall speak for the Oxford and Cambridge Club, in a joint event hosted by Maths and Science Group and the Military History Group, an evening (6 June 2019) with dinner and talks on the theme of the Enigma and Code breaking.

Abstract: I shall describe Alan Turing’s transformative philosophical analysis of the nature of computation, including his argument that some mathematical questions must inevitably remain beyond our computational capacity to answer.

The talk will highlight ideas from Alan Turing’s phenomenal 1936 paper on computable numbers:

This will be a talk for the Theory Seminar for the theory research group in Theoretical Computer Science at Queen Mary University of London. The talk will be held 4 June 2019 1:00 pm, ITL first floor.

Abstract. Curious, often paradoxical instances of self-reference inhabit deep parts of computability theory, from the intriguing Quine programs and Ouroboros programs to more profound features of the Gödel phenomenon. In this talk, I shall give an elementary account of the universal algorithm, showing how the capacity for self-reference in arithmetic gives rise to a Turing machine program $e$, which provably enumerates a finite set of numbers, but which can in principle enumerate any finite set of numbers, when it is run in a suitable model of arithmetic. In this sense, every function becomes computable, computed all by the same universal program, if only it is run in the right world. Furthermore, the universal algorithm can successively enumerate any desired extension of the sequence, when run in a suitable top-extension of the universe. An analogous result holds in set theory, where Woodin and I have provided a universal locally definable finite set, which can in principle be any finite set, in the right universe, and which can furthermore be successively extended to become any desired finite superset of that set in a suitable top-extension of that universe.

This will be a talk at the Institute of Logic, Language and Computation (ILLC) at the University of Amsterdam for events May 11-12, 2019. See Joel David Hamkins in Amsterdam 2019.

Abstract: Potentialism can be seen as a fundamentally model-theoretic notion, in play for any class of mathematical structures with an extension concept, a notion of substructure by which one model extends to another. Every such model-theoretic context can be seen as a potentialist framework, a Kripke model whose modal validities one can investigate. In this talk, I’ll explain the tools we have for analyzing the potentialist validities of such a system, with examples drawn from the models of arithmetic and set theory, using the universal algorithm and the universal definition.

This will be a talk for the Wijsgerig Festival DRIFT 2019, held in Amsterdam May 11, 2019. The theme of the conference is: Ontology.

Abstract. What does it mean to make existence assertions in mathematics? Is there a mathematical universe, perhaps an ideal mathematical reality, that the assertions are about? Is there possibly more than one such universe? Does every mathematical assertion ultimately have a definitive truth value? I shall lay out some of the back-and-forth in what is currently a vigorous debate taking place in the philosophy of set theory concerning pluralism in the set-theoretic foundations, concerning whether there is just one set-theoretic universe underlying our mathematical claims or whether there is a diversity of possible set-theoretic worlds.

This will be talk for the CUNY Set Theory seminar, Friday, March 22, 2019, 10 am in room 6417 at the CUNY Graduate Center.

Abstract. I shall discuss recent joint work with Victoria Gitman and Asaf Karagila, in which we proved that Kelley-Morse set theory (which includes the global choice principle) does not prove the class Fodor principle, the assertion that every regressive class function $F:S\to\text{Ord}$ defined on a stationary class $S$ is constant on a stationary subclass. Indeed, it is relatively consistent with KM for any infinite $\lambda$ with $\omega\leq\lambda\leq\text{Ord}$ that there is a class function $F:\text{Ord}\to\lambda$ that is not constant on any stationary class. Strikingly, it is consistent with KM that there is a sequence of classes $A_n$, each containing a class club, but the intersection of all $A_n$ is empty. Consequently, it is relatively consistent with KM that the class club filter is not $\sigma$-closed.

I am given to understand that the talk will be streamed live online. I’ll post further details when I have them.

Abstract. An old argument, heard perhaps at a good math tea, proceeds: “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.” Does it withstand scrutiny? In this talk, I will 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? Must indiscernible elements in a mathematical structure be automorphic images of one another? We shall discuss many elementary yet interesting 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.

This will be a talk for the Jowett Society on 8 February, 2019. The talk will take place in the Oxford Faculty of Philosophy, 3:30 – 5:30pm, in the Lecture Room of the Radcliffe Humanities building.

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.