The Church of Logic podcast, April 2025

I was interviewed by Cody Roux for The Church of Logic podcast—a fascinating sweeping conversation on issues in the philosophy of mathematics and set theory, including what I described as a fundamental dichotomy between two perspectives on the nature of mathematics and what it is all about. Cody and I have affinities with opposite sides of this dichotomy, which made for a fruitful exchange.

A potentialist conception of ultrafinitism, Columbia University, April 2025


This will be a talk for the conference on Ultrafinitism: Physics, Mathematics, and Philosophy at Columbia University in New York, April 11-13, 2025.

Abstract. I shall argue in various respects that ultrafinitism is fruitfully understood from a potentialist perspective, an approach to the topic that enables certain formal treatments of ultrafinitist ideas, which otherwise often struggle to find satisfactory formalization.

The hierarchy of consistency strengths for membership in a computably enumerable set, Oxford Logic Seminar, May 2025

 This will be a talk for the Logic Seminar at the Mathematical Institute of the University of Oxford, 29 May 2025 5pm Andrew Wiles Building.

Abstract. For a given computably enumerable set $W$, consider the spectrum of assertions of the form $n\in W$. If $W$ is c.e. but not computably decidable, it is easy to see that many of these statements will be independent of PA, for otherwise we could decide $W$ by searching for proofs of $n\notin W$. 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 joint work with Atticus Stonestrom (Notre Dame).

Introduction to modal model theory, Panglobal Algebra and Logic Seminar, Boulder, March 2025

This will be a talk for the Panglobal Algebra and Logic seminar at the University of Colorado Boulder, March 12, 2025, 3:30pm MDT

The talk will be available live on Zoom. Contact the organizers for access.

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 of all models of a fixed first-order theory. In this talk, I shall describe some of the resulting elementary theory, particularly the remarkable expressive power of modal graph theory. This is joint work with my Oxford student Wojciech Wołoszyn.

2025 William Reinhardt Memorial Lecture, Boulder

I am honored to be giving the 2025 William Reinhardt Memorial Lecture at the University of Colorado Boulder, March 11, 2025.

How we might have taken the Continuum Hypothesis as a fundamental axiom, necessary for mathematics

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.

On Skolem’s paradox and the transitive submodel theorem, Rust Belt Workshop in the Philosophy of Logic, Language, and Mathematics, February 2025

This will be a talk for the Rust Belt Workshop in the Philosophy of Logic, Language, and Mathematics, held at Ohio State University in Columbus, Ohio, February 8-9, 2025, University Hall (230 N Oval Mall, Columbus, OH) Room 386B.

Abstract. One can find in the philosophical research literature surrounding Skolem’s paradox a certain claim, referred to as the transitive submodel theorem, according to which every transitive model of set theory admits a countable transitive submodel of the same theory. Although the statement may initially appear quite plausible—perhaps one thinks it follows  from an application of the downward Löwenheim-Skolem theorem—nevertheless it turns out that as a mathematical claim, it is overstated, and there is no such theorem. It is a mistake, although an interesting mistake worth discussing. In this talk I shall give a full account of the countable transitive submodel proposition, taken as a principle of set theory, by showing from suitable hypotheses that counterexamples are possible, characterizing exactly the circumstances in which the principle does hold, and investigating the consistency strength of the proposition and also the consistency strength of its negation. Ultimately, the countable transitive submodel proposition should be seen as a certain anti-large cardinal principle that is equiconsistent with but independent of ZFC, refuted by all the moderately strong large cardinal notions.

This is joint work in progress with Timothy Button, with thanks to W. Hugh Woodin.

I will post a link to the paper when it is available.

The covering reflection principle – Oberwolfach January 2025

This was a talk I gave at the Set Theory Workshop at the Mathematisches Forschungsinstitut in Oberwolfach, Germany, 12-17 January 2025.

Abstract. The principle of covering reflection holds of a cardinal κ if for every structure B in a countable first-order language there is a structure A of size less than κ, such that B is covered by elementary images of A in B. Is there any such cardinal? Is the principle consistent? Does it have large cardinal strength? This is joint work with myself, Nai-Chung Hou, Andreas Lietz, and Farmer Schlutzenberg.

The Human Podcast: 10 questions in 10 minutes

I had an enjoyable little discussion with Joe Murray of The Human Podcast, part of his new series, called 10 questions in 10 minutes, in which he asks his interview subjects for short answers to ten quick questions on their topic. Here is our conversation:

Joe was adamant about the 1 minute timeline for each question, and was holding up timers and giving me the 5 second warning and so forth, but of course, it was simply impossible! There was no way for me to contain my answers to the time limit.

Meanwhile, you can follow through to our previous, longer discussion here:

Every worldly cardinal admits a Gödel-Bernays structure

My Oxford student Emma Palmer and I have been thinking about worldly cardinals and Gödel-Bernays GBC set theory, and we recently came to a new realization.

Namely, what I realized is that every worldly cardinal $\kappa$ admits a Gödel-Bernays structure, including the axiom of global choice. That is, if $\kappa$ is worldly, then there is a family $X$ of sets so that $\langle V_\kappa,\in,X\rangle$ is a model of Gödel-Bernays set theory GBC including global choice.

For background, it may be helpful to recall Zermelo’s famous 1930 quasi-categoricity result, showing that the inaccessible cardinals are precisely the cardinals $\kappa$ for which $V_\kappa$ is a model of second-order set theory $\text{ZFC}_2$.

If one seeks only the first-order ZFC set theory in $V_\kappa$, however, then this is what it means to say that $\kappa$ is a worldly cardinal, a strictly weaker notion. That is, $\kappa$ is worldly if and only if $V_\kappa\models\text{ZFC}$. Every inaccessible cardinal is worldly, by Zermelo’s result. But more, every inaccessible is a limit of worldly cardinals, and so there are many worldly cardinals that are not inaccessible. The least worldly cardinal, for example, has cofinality $\omega$. Indeed, the next worldly cardinal above any ordinal has cofinality $\omega$.

Meanwhile, to improve slightly on Zermelo, we can observe that if $\kappa$ is inaccessible, then $V_\kappa$ is a model of Kelley-Morse set theory when equipped with the full second-order complement of classes. That is, $\langle V_\kappa,\in,V_{\kappa+1}\rangle$ is a model of KM.

This is definitely not true when $\kappa$ is merely worldly and not inaccessible, however, for in this case $\langle V_\kappa,\in,V_{\kappa+1}\rangle$ is never a model of KM nor even GBC when $\kappa$ is singular. The reason is that the singularity of $\kappa$ would be revealed by a short cofinal sequence, which would be available in the full power set $V_{\kappa)+1}=P(V_\kappa)$, and this would violate replacement.

So the question is:

Question. If $\kappa$ is worldly, then can we equip $V_\kappa$ with a suitable family $X$ of classes so that $\langle V_\kappa,\in,X\rangle$ is a model of GBC?

The answer is Yes!

What I claim is that for every worldly cardinal $\kappa$, there is a definably generic well order $\newcommand\slantleq{\mathrel{⩽}}\slantleq$ of $V_\kappa$, so that the subsets definable in $\langle V_\kappa,\in,\slantleq\rangle$ make a model of GBC.

To see this, consider the class forcing notion $\newcommand\P{\mathbb{P}}\P$ for adding a global well order $\slantleq$, as $V_\kappa$ sees it. Conditions are well orders of some $V_\alpha$ for some $\alpha<\kappa$, ordered by end-extension, so that lower rank sets always preceed higher rank sets in the resulting order.

I shall prove that there is a well-order $\slantleq$ that is generic with respect to dense sets definable in $\langle V,\in\rangle$.

For this, let us consider first the case where the worldly cardinal $\kappa$ has countable cofinality. In this case, we can find an increasing sequence $\kappa_n$ cofinal in $\kappa$, such that each $\kappa_n$ is $\Sigma_n$-correct in $V_\kappa$, meaning $V_{\kappa_n}\prec_{\Sigma_n}V_\kappa$.

In this case, we can build a definably generic filter $G$ for $\P$ in a sequence of stages. At stage $n$, we can find a well order up to $\kappa_n$ that meets all $\Sigma_n$ definable dense classes using parameters less than $V_{\kappa_n}$. The reason is that for any such definable dense set, we can meet it below $\kappa_n$ using the $\Sigma_n$-correctness of $\kappa_n$, and so by considering various parameters in turn, we can altogether handle all parameters below $V_{\kappa_n}$ using $\Sigma_n$ definitions. That is, the $n$th stage is itself an iteration of length $\kappa_n$, but it will meet all $\Sigma_n$ definable dense sets using parameters in $V_{\kappa_n}$.

Next, we observe that the ultimate well-order of $V_\kappa$ that arises from this construction after all stages is fully definably generic, since any definition with arbitrary parameters in $V_\kappa$ is a $\Sigma_n$ definition with parameters in $V_{\kappa_n}$ for some large enough $n$, and so we get a definably generic well order $\slantleq$. Therefore, the usual forcing argument shows that we get GBC in the resulting model $\langle V_\kappa,\in,\text{Def}(V_\kappa)\rangle$, as desired.

The remaining case occurs when kappa has uncountable cofinality. In this case, there is a club set $C\subseteq\kappa$ of ordinals $\gamma\in C$ with $V_\gamma\prec V_\kappa$. (We can just intersect the clubs $C_n$ of the $\Sigma_n$-correct cardinals.) Now, we build a well-order of $V_\kappa$ that is definably generic for every $V_\gamma$ for $\gamma\in C$. At limits, this is free, since every definable dense set in V_lambda with parameters below is also definable in some earlier $V_\gamma$. So it just reduces to the successor case, which we can get by the arguments above (or by induction). The next correct cardinal $\gamma$ above any ordinal has countable cofinality, since if one considers the next $\Sigma_1$-correct cardinal, the next $\Sigma_2$-correct cardinal, and so on, the limit will be fully correct and cofinality $\omega$.

The conclusion is that every worldly cardinal $\kappa$ admits a definably generic global well-order on $V_\kappa$ and therefore also admits a Gödel-Bernays GBC set theory structure $\langle V_\kappa,\in,X\rangle$, including the axiom of global choice.

The argument relativizes to any particular amenable class $A\subseteq V_\kappa$. Namely, if $\langle V_\kappa,\in,A\rangle$ is a model of $\text{ZFC}(A)$, then there is a definably generic well order $\slantleq$ of $V_\kappa$ such that $\langle V_\kappa,\in,A,\slantleq\rangle$ is a model of $\text{ZFC}(A,\slantleq)$, and so by taking the classes definable from $A$ and $\slantleq$, we get a GBC structure $X$ including both $A$ and $\slantleq$.

This latter observation will be put to good use in connection with Emma’s work on the Tarski’s revenge axiom, in regard to finding the optimal consistency strength for one of the principles.

The computable surreal numbers, Notre Dame Logic Seminar, December 2024

This will be a talk for the Notre Dame Logic Seminar, 3 December 2024, 2:00pm, 125 Hayes-Healey.

Abstract. I shall give an account of the theory of computable surreal numbers, proving that these form a real-closed field. Which real numbers arise as computable surreal numbers? You may be surprised to learn that some noncomputable real numbers have computable surreal presentations, and indeed the computable surreal real numbers are exactly the hyperarithmetic reals. More generally, the computable surreal numbers are exactly those with a hyperarithmetic surreal sign sequence. This is joint work with Dan Turetsky, but we subsequently found that it is a rediscovery of earlier work of Jacob Lurie.

Lecture notes:

See related MathOverflow posts:

Also see my elementary introduction to the surreal numbers: The surreal numbers

Determinateness of truth does not come for free from determinateness of objects, Singapore, November 2024

 This will be a talk for the (In)determinacy in Mathematics conference at the National University of Singapore, 20-22 November 2024

Abstract. I shall discuss the question whether we may regard determinateness of truth as flowing from determinateness of objects in a mathematical structure. I shall showcase several results in the model theory of set theory and arithmetic that seem to speak against this. For example, there are two models of ZFC set theory that share exactly the same arithmetic structure of the natural numbers ⟨ℕ,+,·,0,1,<⟩, what they each view as the standard model of arithmetic, but they disagree about which arithmetic sentences are true in that structure. There are models of ZFC set theory with the same arithmetic structure and the same arithmetic truth, but which disagree on truth-about-truth, or that agree on that, but disagree on higher levels of iterated truth, at any desired level. There are models of set theory with the same natural numbers and real numbers, but which disagree on projective truth. There are models of ZFC that have a rank initial segment Vθ in common, but they disagree about whether it is a model of ZFC. All these examples show senses in which determinateness about objects does not seem to cause determinateness about truth. (This is joint work with Ruizhi Yang.)

The covering reflection theorem, Madison Logic Seminar, October 2024

This will be a talk at the UW Madison Logic Seminar on 22 October 2024.

Abstract. The principle of covering reflection holds of a cardinal κ if for every structure B in a countable first-order language there is a structure A of size less than κ, such that B is covered by elementary images of A in B. Is there any such cardinal? Is the principle consistent? Does it have large cardinal strength? This is joint work with myself, Nai-Chung Hou, Andreas Lietz, and Farmer Schlutzenberg.

The talk will reportedly streamed online, so kindly contact the organizers for access.

I will be staying in Madison for a few days to talk logic with researchers there.

Infinite-time computable analogues of the universal algorithm, Generalized Computability Theory Workshop, Spain, August 2024

This will be a talk at the Generalized Computability Theory workshop in Castro Urdiales, Spain, a beautiful setting on the sea near Bilbao, 19-23 August 2024.

Abstract. I shall present infinite-time computable analogues of the universal algorithm, which can in principle produce any desired output stream, if only it is run in the right set-theoretic universe, and then extended as desired in further universes.

How we might have viewed the continuum hypothesis as a fundamental axiom necessary for mathematics, Oxford Phil Maths seminar, May 2025

This will be a talk for the Philosophy of Mathematics Seminar at the University of Oxford, 19 May 2025.

Abstract. I shall describe a simple historical thought experiment showing how our attitude toward the continuum hypothesis could 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 view the continuum hypothesis as a fundamental axiom of set theory, necessary for mathematics and indeed indispensable for calculus.

I shall be speaking on my paper: How the continuum hypothesis could have been a fundamental axiom

Puzzles of reality and infinity, Mindscape Podcast

I was interviewed by Sean Carroll for his Mindscape Podcast, broadcast 15 July 2024.