Category Archives: Talks

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

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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:

Lecture Notes – Computable surreal numbers ND Dec 2024Download

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

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 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.)

Slides – Determinateness of truth – Singapore 2024Download

The covering reflection theorem, Madison Logic Seminar, October 2024

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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.

Slides – The covering reflection principle – Madison Oct 2024Download

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

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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.

Slides – Infinitary universal algorithm – Spain 2024 – HamkinsDownload

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

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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

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

282 | Joel David Hamkins on Puzzles of Reality and Infinity

Mathematics, Philosophy of Set Theory and Infinity, Back to the Stone Age interview, May 2024

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I was interviewed by Francesco Cavina for the Back to the Stone Age series on May 17, 2024, with a sweeping discussion of the philosophy of set theory, infinity, the continuum hypothesis, beauty in mathematics, and much more.

Life Story of Mathematician & Philosopher of Infinity, interviewed by The Human Podcast, May 2024

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I was interviewed by The Human Podcast on 17 May 2024. Please enjoy our sweeping conversation about nature of infinity, the nature of abstract mathematical existence, the applicability of mathematical abstractions to physical reality, and more. At the end, you will see that I am caught completely at a loss in answer to the question, “What is it to live a good life?”.

Forcing is simply the iterative conception undertaken with multivalued logic, ForcingFest, Oslo, June 2024

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I shall be speaking at the ForcingFest meeting at the University of Oslo, 21 June 2024.

Abstract. I will explain how the forcing construction can be seen as a direct implementation of the iterative conception, giving rise to the cumulative hierarchy, but undertaken in the context of multivalued logic. The shape of the logic that is available in effect enables a certain constructive interference of the truth values in such a way that can affect the truth judgements. The core utility of forcing arises from the fact that we can often control these consequences by making a careful choice of the logic to be used, thereby controlling the truth values even of natural set-theoretic statements such as the continuum hypothesis.

Slides-ForcingFest-Oslo-Hamkins-June-2024Download

The continuum hypothesis could have been a fundamental axiom, CFORS Grad Conference, Oslo, June 2024

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I shall be giving a keynote lecture for the CFORS Grad Conference at the University of Oslo, 19-20 June 2024.

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, one furthermore necessary for mathematics and indeed, indispensable for calculus.

Slides-CH-could-have-been-fundamental-Hamkins-Oslo-June-2024Download

The paper is now available at How the continuum hypothesis could have been a fundamental axiom.

Failing definite descriptions, Notre Dame Food for Thought Seminar, March 2024

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I gave a talk for the Food for Thought seminar for the Notre Dame philosophy department.

The topic concerned definite descriptions, particularly the semantics that might be given when one extends first-order logic to include the iota operator, by which $℩x\varphi(x)$ means “the $x$ such that $\varphi(x)$.” There are a variety of natural ways to define the semantics of iota assertions in a model, and we discussed the advantages and disadvantages of each approach. We concentrated on what I call the strong semantics, the weak semantics, and the natural semantics, respectively. Ultimately, I argue for a deflationary perspective on the debate, as each of the semantics is conservative over the base language, with no iota operator, with no new expressive power. In this sense, I argue, the choice of one semantics over another is purely a matter of convenience or ease of expressibility, as all of the notions are expressible without definite descriptions at all.

My lecture notes are below.

Failing-definite-descriptionsDownload

How the continuum hypothesis could have been a fundamental axiom, UC Irvine Logic & Philosoph of Science Colloquium, March 2024

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This will be a talk for the Logic and Philosophy of Science Colloquium at the University of California at Irvine, 15 March 2024.

Abstract. With a simple historical thought experiment, I should like to describe how we might easily have come to view the continuum hypothesis as a fundamental axiom, one necessary for mathematics, indispensable even for calculus.

Slides-CH-could-have-been-fundamental-Hamkins-Irvine-March-2024Download

The paper is now available at How the continuum hypothesis could have been a fundamental axiom.

What if your potentialism is implicitly actualist? Oxford conference, March 2024

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This will be a talk at the conference Challenging the Infinite, March 11-12 at Oxford University. (Please register now to book a place.)

Abstract Many commonly considered forms of potentialism, I argue, are implicitly actualist in the sense that a corresponding actualist ontology and theory is interpretable within the potentialist framework using only the resources of the potentialist ontology and theory. And vice versa. For these forms of potentialism, therefore, there seems to be little at stake in the debate between potentialism and actualism—the two perspectives are bi-interpretable accounts of the same underlying semantic content. Meanwhile, more radical forms of potentialism, lacking convergence and amalgamation, do not admit such a bi-interpretation with actualism. In light of this, the central dichotomy in potentialism, to my way of thinking, is not concerned with any issue of height or width, but rather with convergent versus divergent possibility.

Slides-What-if-your-potentialism-is-implicitly-actualist-Oxford-March-2024Download

The covering reflection principle, Notre Dame Logic Seminar, February 2024

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This will be a talk for the Notre Dame Logic Seminar on 6 February 2024, 2:00 pm.

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

Covering-reflection-ND-Logic-SeminarDownload

The Gödel incompleteness phenomenon, interview with Rahul Sam

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Please enjoy my conversation with Rahul Sam for his podcast, a sweeping discussion of topics in the philosophy of mathematics—potentialism, pluralism, Gödel incompleteness, philosophy of set theory, large cardinals, and much more.