Recently I had the pleasure to give a talk at the Conference on the occasion of Jörg Brendle’s 60th birthday at Kobe University in Japan, and I was invited to make remarks at the conference banquet given in his honor.

I feel a special affinity with Jörg, since we had both first come to Japan in 1998 at almost exactly the same time, within a week of each other, and naturally we were faced with similar bewildering differences in language and culture in those first early days after arriving in Japan. I had taken up a one-year JSPS Research Fellowship at Kobe University, while Jörg had entered into his permanent position there. He went on to build an impressive research career there with many accomplishments and many students. My own stay in Japan was a formative period in my life, greatly enjoyed.

At the banquet, I related some anecdotes from the early days, but also expressed heartfelt philosophical reflections on the significance of major life decisions and the sanguineity one might feel about alternative life paths not taken. What a blissful enjoyable life I might have had, for example, if I had somehow managed to remain in Japan and build a life there as Jörg has done.

I proceed to continue my remarks as follows regarding the Sannoymiya incident...

Allow me, finally, to relate what I know of the infamous Sannomiya incident, an event which marks to my way of thinking the time when Jörg first made his big splash in the Japanese art world, the time when he first hit the big stage.

It was a late summer afternoon, early evening, and Jörg, wearing his customary straw hat, was walking home down the mountain from the university into town. Carrying some newly procured oil paints, he was full of anticipation and artistic plans. Inadvisably, however, in his excitement he opened some of the tubes while he was walking, in order to judge the quality, in part by smelling the oils. At this, he was startled by a coarse grunt behind him on the trail—an inoshishi* had noticed the smell of the oils and begun to follow Jörg. And it wasn’t one of those small inoshishi that you can scare away by grunting yourself, but rather a large bold inoshishi. Jörg nervously quickened his pace to get away, gripping the oils tightly, but the inoshishi was gaining on him.

In his preoccupation with the inoshishi, Jörg hadn’t noticed the large crane flying over head. But the crane had definitely noticed Jörg, or more specifically, his straw hat, which I suppose the bird coveted for his nest. The bird swooped down and, with surprising speed, grabbed the hat directly off Jörg’s head!

Of course Jörg was not going to let a bird steal his hat, and he held fast onto it. In the struggle, with the bird’s great wings flapping and Jörg holding tight to the hat, he was stretched upward. The inoshishi saw his chance and charged right between Jörg’s legs, who was thrust up onto the back of the inoshishi, finding himself to be riding it like a horse.

The three of them were set galloping down the mountain. The great crane ahead, flapping its wings and pulling the hat, to which Jörg clung, while riding the squealing inoshishi, who carried them all down the mountain trail at full speed. Imagine the sight. Caught in the tight grasp of Jörg’s hands, paint began to ooze out of the tubes, which began to make a mess on Jörg, dripping also onto the inoshishi, who became increasingly splotched with various bright colors.

The trail came down to the precipice, where normally one would make the turn and continue on the trail downhill. But the crane had no reason to turn, and instead flew straight out over the edge. Inexplicably, Jörg did not let go and found himself dangling precariously as the bird flew out over the neighborhood below. The inoshishi, wrapped in Jörg’s legs and taken completely by surprise with this new situation, was gripping for dear life on to Jörg. In the frantic struggle, the inoshishi had bitten off both of Jörg’s shoes, which fell away below.

Thus the absurd triple—the great flying crane holding the hat, with now barefoot Jörg dangling below and the squealing inoshishi, covered with paint—soared over the city, heading to Sannomiya in the Kobe city center.

At that time there was a Matsuri summer festival taking place downtown, and people strolled about in elegant yukatas*, browsing the various fine fabrics on display, the theme of the festival. A great linen fabric, with an almost imperceptible white on white pattern, was spread across the big stage at the center of the festival, surrounded by onlookers who stood around it taking in its subtle, starkly spare nature.

Presently all attention turned skyward—the festival crowd gasped with bewilderment as Jörg and the crane and the squealing inoshishi came into view overhead. By this time, of course, the crane was tiring, and suddenly it just let go. The tangled paint-smeared mess of Jörg and the inoshishi fell to Earth, Splash!, landing right on the great white linen canvas spread upon the big stage.

The inoshishi stood up, grunting and squealing, but then, looking at the spot of the fall, let off a strange twitch and became suddenly transfixed, relaxed and silent. The crowd, initially full of shouts and alarm, also became suddenly calm, every person looking intently before them. Stepping back in his awkward barefootedness, Jörg realized that the people were not looking at him or at the inoshishi at all, but rather directly at the fabric, right at the spot right where they had landed. The inoshishi seemed as though hypnotized by the canvas, of course now smeared with paint from the crash, overcome with its......beauty. The people were similarly dumbstruck, overcome with emotion and profound meaning—to a person they gazed at the sublime work of art that Jörg and the inoshishi had created on the canvas.

A voice broke through the silence Ichi man en!*, an opening bid in the hotly contested impromptu auction that ensued. Soon, the work was sold for a great sum. Jörg was asked to sign his mark.

So that is what I know of the story of the Sannomiya incident, in which Jörg first made his big splash in the Japanese art scene, the time when he hit the big stage.

Now, I have heard that some people doubt this story, but then again, some people doubt the existence of a unique absolute set-theoretic universe, so evidently like Descartes one can doubt absolutely anything, even incontrovertible truths such as the account I have just related.

Joel David Hamkins
4 September 2025
Kobe, Japan

*An inoshishi is a kind of wild boar, commonly seen in the early evenings in the Kobe hills, including the university campus

*A yukata is a traditional cotton summer kimono, often worn by both men and women at summer festivals

*The phrase “ichi man en” means ten-thousand yen

Art Show next week

Incidentally, Jörg has an art show exhibition in Japan next week. Follow the links to the main show, which lists a few of his pieces there. Look under the category “Award Winning”.

• https://shunyo-kai.or.jp/lists-2/list-101/
• https://shunyo-kai.or.jp/lists-2/list-102/

Mathematician’s year in Japan

For those who are interested, I wrote a book A mathematician’s year in Japan relating my experiences in Japan during that year I lived in Kobe.

This will be a talk for the Conference on the occasion of Jörg Brendle’s 60th birthday at Kobe University in Kobe, Japan, 2-5 September 2025.

Many years ago, I was a JSPS Fellow at Kobe University, at the same time that Jörg first took up his position in Japan, a time when Philip Welch also had his professorship there.

Title The elementary theory of surreal arithmetic is bi-interpretable with set theory

Speaker Joel David Hamkins, O’Hara Professor of Logic, University of Notre Dame

Abstract I shall introduce the first-order elementary theory of surreal arithmetic, a theory that is true in the surreal field when equipped with its birthday order. This structure is bi-interpretable with the set-theoretic universe (V,∈), and indeed the theory of surreal arithmetic SA is bi-interpretable with ZFC. This is a preliminary report on very new joint work in progress with myself, Junhong Chen, and Ruizhi Yang, both of Fudan University, Shanghai.

This will be a talk I shall give for the History and Philosophy of Science (HPS) Colloquium at the University of Notre Dame, 17 October 2025, 12:30-1:30 pm, 201 O’Shaughnessy Hall.

Did Turing ever halt?

Abstract. Alan Turing’s 1936 paper on computable numbers, perhaps one of the most impactful papers ever written, arguably spawned the fields of computability theory, complexity theory, and computer science, helping to usher in the computer age. He introduced Turing machines, provided the first universal computers, launched the investigation of the computable numbers, and proved the first instances of computable undecidability. Turing 1936 is widely credited, in nearly all the standard computability textbooks, for the undecidability of the halting problem, often viewed today as the seminal undecidability result, leading to all the others. The curious historical situation, however, is that there is no mention at all of the halting problem in Turing’s article and in fact Turing never considers the halting of his machines—he specifically designed them to run forever. In this talk (joint work with Theodor Nenu, Oxford), I shall discuss the curious history of the halting problem and the question of whether we rightly credit the undecidability result to Turing. I shall come eventually to a nuanced conclusion.

See the relevant paper:

Joel David Hamkins and Theodor Nenu, "Did Turing prove the undecidability of the halting problem?", 18 pages, 2024, Mathematics arXiv:2407.00680.

This will be a talk for the Fudan Logic Seminar at Fudan University, to be followed immediately by two talks for the Fudan Logic student seminar, forming a mini-conference for the logic group on 23 July 2025.

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 are computable as surreal numbers, 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 in part of earlier work of Jacob Lurie.

This will be a talk for the Seminar on Frontier Issues in Logic and Philosophy

The First Forum on Logic and Philosophy

逻辑与哲学前沿问题研究"学术研讨会暨第一届逻辑与哲学论坛

Changchun, China, 18-20 July 2025

Pointwise definable end-extensions of models of arithmetic and set theory

Abstract. The existence of pointwise definable models of set theory offers a fundamental engagement with what has become known as the Math Tea argument, according to which there must be undefinable real numbers, since there are only countable many definitions, but uncountably many real numbers. I shall present a new flexible model-theoretic method showing that every countable model of Peano Arithmetic (PA) admits a pointwise definable end-extension, one in which every object is definable without parameters. The argument makes a fundamental use of the universal algorithm. 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 lecture series on the Philosophy of Mathematics at Fudan University in Shanghai, China, 30 June – 25 July 2025, as a part of the International Summer School program at Fudan University. Lectures given by Ruizhi Yang and myself.

My lectures begin 7 July, with the following themes:

• Numbers
• Infinity
• Geometry
• Proof
• Computability
• Incompleteness
• Set theory

Students may find my treatment of these themes in my book, Lectures on the Philosophy of Mathematics, to be a helpful resource.

This will be a talk for the International Conference on the Philosophy of Mathematics, held at Lanzhou University, China, 25-27 July 2025.

How the continuum hypothesis might have been a fundamental axiom

Abstract. I shall describe a 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.

See related paper: How the continuum hypothesis could have been a fundamental axiom

This will be a talk for the Conference on Infinity, a collaborative meeting of logicians and specialists in Chinese philosophy here at Peking University, 24 June 2025, in the philosophy department.

Abstract. I shall lay out a spectrum of fundamentally different potentialist conceptions of infinity. The differences in these potentialist ideas become especially clear when adopting a modal perspective on potentialism, grounded in the ideas of modal logic. I shall argue that some forms of potentialism, the “convergent” forms, are implicitly actualist, whereas the radical branching form of potentialism is more truly potentialist in nature.

This will be a lecture series at Peking University in Beijing in June 2025.

Announcement at Peking University

Course abstract. This will be a series of advanced lectures on set theory, treating diverse topics and particularly those illustrating how set theoretic ideas and conceptions shed light on core foundational matters in mathematics. We will study the pervasive independence phenomenon over the Zermelo-Fraenkel axioms of set theory, perhaps the central discovery of 20th century set theory, as revealed by the method of forcing, which we shall study in technical detail with numerous examples and applications, including iterated forcing. We shall look into all matters of the continuum hypothesis and the axiom of choice. We shall introduce the basic large cardinal axioms, those strong axioms of infinity, and investigate the interaction of forcing and large cardinals. We shall explore issues of definability and truth, revealing a surprisingly malleable nature by the method of forcing. Looking upward from a model of set theory to all its forcing extensions, we shall explore the generic multiverse of set theory, by which one views all the models of set theory as so many possible mathematical worlds, while seeking to establish exactly the modal validities of this conception. Looking downward in contrast transforms this subject to set-theoretic geology, by which one understands how a given set-theoretic universe might have arisen from its deeper grounds by forcing. We shall prove the ground-model definability theorem and the other fundamental results of set-theoretic geology. The lectures will assume for those participating a certain degree of familiarity with set-theoretic notions, including the basics of ZFC and forcing.

There will be ten lectures, each a generous 3 hours.

Lecture 1. Set Theory

This first lecture begins with fundamental notions, including the dramatic historical developments of set theory with Cantor, Frege, Russell, and Zermelo, and then the rise of the cumulative hierarchy and the iterative conception. The move to a first-order foundational theory. The Skolem paradox. The omission of urelements and the move to a pure set theory. We will establish the reflection phenomenon and the phenomenon of correctness cardinals, before providing some simple relative consistency results. We will compare the first-order approach to the various class theories and also lay out the spectrum of weak theories, including locally verifiable set theory, before discussing countabilism as an approach to set theory.

Lecture 2. Categoricity and the small large cardinals

We will discuss the central role and importance of categoricity in mathematics, highlighting this with results of Dedekind and Huntington, and with several examples of internal categorcity. Afterwards, we shall begin to introduce various small large cardinal notions—the inaccessible cardinals, the hyperinaccessibility hierarchy, Mahlo cardinals, worldly cardinals, other-worldly cardinals. We shall explain the connection with categoricity via Zermelo’s categoricity result. Going deeper, we discuss the possibility of categorical large cardinals and the enticing possibility of a fully categorical set theory.

Lecture 3. Forcing

We shall give an introduction to forcing, pursuing and comparing two approaches, via partial orders versus Boolean algebras. Forcing arises naturally from the iterative conception of the cumulative hierarchy, when undertaken in multi-valued logic. We shall see the principal introductory forcing examples, including the forcing to add a Cohen real, cardinal collapse forcing, forcing the failure of CH, forcing to add dominating reals, almost disjoint coding, iterated forcing, the forcing of Martin’s axiom, and the case of Suslin trees.

Lecture 4. Continuum Hypothesis

We tell the story of the continuum hypothesis, from Cantor’s initial conception and strategy, to Gödel’s proof of CH in the constructible universe, and ultimately Cohen’s forcing of ¬CH, establishing independence over ZFC. The CH is a forcing switch. We discuss the generalized continuum hypothesis GCH, and prove Easton’s theorem on the continuum function. Finally, we discuss various philosophical approaches to settling the CH problem, including Freiling’s axiom and the equivalence with ¬CH, and the role of the continuum hypothesis in providing a categorical theory of the hyperreals. Two equivalent formulations of CH in ZFC are not equivalent without AC.

Lecture 5. Axiom of Choice

We tell the story of the axiom of choice, beginning with a spectrum of equivalent formulations, including the linearity of cardinality. We discuss the abstract cardinal-assignment problem versus the cardinal-selection problem. We establish the truth of the axiom of choice in the constructible universe, as well as global choice, but ultimately the independence of the axiom of choice over ZF via forcing and the symmetric model construction method. Finally, we discuss the perfect predictor theorem and the box puzzle conundrum.

Lecture 6. Definability

We shall define and discuss the formal notion of definability in mathematics and set theory. Can every set be definable? We exhibit the phenomenon of pointwise definable models and their relevance for the Math Tea argument. We define the inner model HOD and explore its interaction with forcing, forcing V=HOD and also forcing V≠HOD. We reveal the coquettish nature of HOD, establishing the nonabsoluteness of HOD, showing furthermore that every model of set theory is the HOD of another model. We show how forcing generic filters can be definable in their forcing extensions. Finally, we shall exhibit a spectrum of paradoxical examples revealing various subtleties in the notion of definability.

Lecture 7. Truth

What is truth? We establish Tarski’s theorem on the nondefinability of truth, and establish the second incompleteness theorem via the Grelling-Nelson paradox. We analyze the connection between truth predicates and correctness cardinals. What is the consistency strength of having a truth predicate? Can a model of set theory contain its own theory as an element? Must it? We define the truth telling game. We shall force a definable truth predicate for HOD. We shall establish the nonabsoluteness of satisfaction.

Lecture 8. Forcing and large cardinals

Can large cardinals settle CH? Gödel had hoped so, but this is refuted by the Levy-Solovay theorem. We will prove forcing preservation theorems for large cardinals, and nonabsoluteness theorems. On the difference between lifting and extending measures. Laver indestructibility and the lottery preparation, via master condition arguments.

Lecture 9. Set-theoretic geology

Looking down, we shall give an introduction to set-theoretic geology. We will prove the ground model definability theorem, using the cover and approximation properties. We shall define the Mantle and prove that every model of set theory is the Mantle of another model. We will discuss Bukovski’s theorem characterizing forcing extensions and prove Usuba’s theorems on the downward directedness of grounds.

Lecture 10. Set-theoretic potentialism

Looking up, we view forcing as a modality, viewing every model of set theory in the context of its generic multiverse. We shall investigate the modal logic of forcing with independent buttons and switches. We shall explore the other natural interpretations of set-theoretic potentialism and investigate their modal validities.

Comments or suggestions welcome.

This will be an online talk for the Leeds Set Theory Seminar, 21 May 2025 1pm BST. Contact the organizers (Hope Duncan) for Teams access.

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 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. There is no such theorem. In this talk I shall give a full account of the countable transitive submodel proposition, taken as a principle of set theory, showing from suitable hypotheses that counterexamples are possible and characterizing exactly the circumstances in which the principle does hold. Ultimately, the countable transitive submodel proposition should be seen as a certain anti-large cardinal principle that is equiconsistent with but independent of ZFC and 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 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.

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.

Handout format, without pauses: Slides – Ultrafinitism – Columbia 2025 – Hamkins – handout

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,ドル 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).

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.

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.