Category Archives: Talks

Varieties of potentialism, Oslo, April 2023

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This will be an online talk for the Infinity & Intentionality project of Øystein Linnebo in Oslo, 25 April 2023. Zoom link available from the organizers.

Abstract: I shall survey the surprisingly enormous variety of potentialist conceptions, even in the case of arithmetic potentialism, spanning a spectrum from linear inevitabilism and other convergent potentialist conceptions to more radical nonamalgamable branching-possibility potentialist conceptions. Underlying the universe-fragment framework for potentialism, one finds a natural modal vocabulary capable of expressing fine distinctions between the various potentialist ideas, as well as sweeping potentialist principles. Similarly diverse conceptions of ultrafinitism grow out of the analysis. Ultimately, the various convergent potentialist conceptions, I shall argue, are implicitly actualist, reducing to and interpreting actualism via the potentialist translation, whereas the radical-branching nonamalgamable potentialist conception admits no such reduction. 

Slides-Varieties-of-potentialism-Hamkins-Oslo-April-2023

Natures of Proof, Pacific APA, San Francisco, April 2023

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I am a commentator at the Pacific APA 2023 conference in San Francisco in a Book Symposium session focused on the book of Catarina Duthil Novaes, The Dialogical Roots of Deduction.

I think very highly of Novaes’s book (my book review is here) and I nominated it for the Lakatos prize, which I am very glad to say that she won. This is particularly appropriate in my view in light of Lakatos’s own use of dialogues in expressing his perspectives on the philosophy of mathematics and the nature of proof.

I shall be a speaker at the book symposium, intending to place the dialogical perspective on proof in the context of a variety of other views of proof. I shall conclude with a few criticisms of the book, which I hope might lead to interesting discussion.

My remarks will follow this outline:

Natures-of-proofDownload

Pointwise definable and Leibnizian extensions of models of arithmetic and set theory, Madison Logic Seminar, April 2023

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Abstract. I shall present a new flexible method showing that every countable model of PA admits a pointwise definable-elementary end-extension. 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.

UW Madison Logic Seminar, Joel David Hamkins, April 4, 2023
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The Math Tea argument: must there be numbers we can neither describe nor define? Barcelona March 2023

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This will be a talk 15 March 2023 for the Mathematics Department of the University of Barcelona, organized jointly with the Set Theory Seminar.

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, including every real number, every
function, every set. We shall discuss the relevance for the math tea argument.

Slides-Pointwise-Definability-Barcelona-2023

A survey of set-theoretic geology, Notre Dame Logic Seminar, January 2023

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This will be a talk 31 January 2-3 for the Notre Dame Logic Seminar.

Abstract. I shall give a general introduction and account of the main elements of set-theoretic geology, the motivating questions, the central definitions, and the main results, including newer advances. We’ll discuss ground models, the ground axiom, the mantle, the ground-model definability theorem, Usuba’s results on downward directedness and more. 

Paradox, Infinity, & The Foundations of Mathematics, interview with Robinson Erhardt, January 2023

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This was an interview with Robinson Erhardt on Robinson’s Podcast, part of his series of interviews with various philosophers, including many philosophers of mathematics and more.

We had a wonderfully wide-ranging discussion about the philosophy of mathematics, the philosophy of set theory, pluralism, and many other topics. The main focus was the topic of infinity, following selections from my new book, The Book of Infinity, currently being serialized on my substack, joeldavidhamkins.substack.com, with discussion of Zeno’s paradox, the Chocolatier’s Game, Hilbert’s Grand Hotel and more.

Robinson compiled the following outline with links to special parts of the interview:

  • 00:00 Introduction
  • 2:52 Is Joel a Mathematician or a Philosopher?
  • 6:13 The Philosophical Influence of Hugh Woodin
  • 10:29 The Intersection of Set Theory and Philosophy of Math
  • 16:29 Serializing the Book of the Infinite
  • 20:05 Zeno of Elea, Continuity, and Geometric Series
  • 39:39 Infinite Games and the Chocolatier
  • 53:35 Hilbert’s Hotel
  • 1:10:26 Cantor’s Theorem
  • 1:31:37 The Continuum Hypothesis
  • 1:43:02 The Set-Theoretic Multiverse
  • 2:00:25 Berry’s Paradox and Large Numbers
  • 2:16:15 Skolem’s Paradox and Indescribable Numbers
  • 2:28:41 Pascal’s Wager and Reasoning Around Remote Events
  • 2:49:35 MathOverflow
  • 3:04:40 Joel’s Impeccable Fashion Sense

Read the book here: joeldavidhamkins.substack.com.

Strategic thinking in infinite games, CosmoCaixa Science Museum, Barcelona, March 2023

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I am deeply honored to be invited by la Caixa Foundation to give a talk in “The Greats of Science” talk series, to be held 16 March 2023 at the CosmoCaixa Science Museum in Barcelona. This talk series aspires to host “prestigious figures who have contributed towards admirable milestones, studies or discoveries,” who will bring the science to a general audience, aiming to “give viewers the chance to explore the most relevant parts of contemporary sicence through the top scientists of the moment.” Previous speakers include Jane Goodall and nearly a dozen Nobel Prize winners since 2018.

Cosmo Caixa announcement

I hope to rise to those high expectations!

My topic will be: Strategic thinking in infinite games.

Have you time for an infinite game? Many familiar finite games admit natural infinitary analogues, infinite games that may captivate and challenge us with intriguing patterns and sublime complexity. Shall we have a game of infinite chess? Or how about infinite draughts, infinite Hex, infinite Wordle, or infinite Sudoku? In the Chocolatier’s game, the Chocolatier serves up an infinite stream of delicious morsels, while the Glutton aims to eat every one. These games and others illustrate the often subtle strategic aspects of infinite games, and sometimes their downright logical peculiarity. Does every infinite game admit of a winning strategy? Must optimal play be in principle computable? Let us discover the fascinating nature of infinitary strategic thinking.

The theory builds upon the classical finitary result of Zermelo (1913), the fundamental theorem of finite games, which shows that in every finite two-player game of perfect information, one of the players must have a winning strategy or both players have draw-or-better strategies. This result extends to certain infinitary games by means of the ordinal game-value analysis, which assigns transfinite ordinal values $\alpha$ to positions in a game, generalizing the familiar mate-in-$n$ idea of chess to the infinite. Current work realizes high transfinite game values in infinite chess, infinite draughts (checkers), infinite Go, and many other infinite games. The highest-known game value arising in infinite chess is the infinite ordinal $\omega^4$, and every countable ordinal arises in infinite draughts, the optimal result. Games exhibiting high transfinite ordinal game values have a surreal absurd character of play. The winning player will definitely win in finitely many moves, but the doomed losing player controls the process with absurdly long deeply nested patterns of forcing moves that must be answered, as though counting down from the infinite game value—when 0 is reached, the game is over.

Infinite-games-Barcelona-CosmoCaixa-2023

Pointwise definable and Leibnizian extensions of models of arithmetic and set theory, MOPA seminar CUNY, November 2022

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 This will be an online talk for the MOPA Seminar at CUNY on 22 November 2022 1pm. Contact organizers for Zoom access.

Abstract. I shall introduce a flexible new method showing that every countable model of PA admits a pointwise definable end-extension, one in which every individual is definable without parameters. And similarly for models of set theory, in which one may also achieve the Barwise extension result—every countable model of ZF admits a pointwise definable end-extension to a model of ZFC+V=L, or indeed any theory arising in a suitable inner model. A generalization of the method shows that every model of arithmetic of size at most continuum admits a Leibnizian extension, and similarly in set theory. 

Slides-Pointwise-definable-end-extensions-MOPA-2022

Pointwise definable and Leibnizian models of arithmetic and set theory, realized in end extensions of a given model, Notre Dame Logic Seminar, October 2022

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This will be a talk for the Notre Dame logic seminar, 11 October 2022, 2pm in Hales-Healey Hall.

Abstract.  I shall present very new results on pointwise definable and Leibnizian end-extensions of models of arithmetic and set theory. Using the universal algorithm, I shall present a new flexible method showing that every countable model of PA admits a pointwise definable $\Sigma_n$-elementary end-extension. 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.

The math tea argument—must there be numbers we cannot describe or define? Pavia Logic Seminar

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This will be a talk for the Philosophy Seminar at the IUSS, Scuola Universitaria Superiore Pavia, 28 September 2022.

(Note: This seminar will be held the day before the related conference Philosophy of Mathematics: Foundations, Definitions and Axioms, Italian Network for the Philosophy of Mathematics, 29 September to 1 October 2022. I shall be speaking at that conference on the topic, Fregean abstraction in set theory, a deflationary account.)

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.

Slides-Pointwise-Definability-Pavia-2022Download

Workshop on the Set-theoretic Multiverse, Konstanz, September 2022

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Masterclass of “The set-theoretic multiverse” ten years after

Focused on mathematical and philosophical aspects of the set-theoretic multiverse and the pluralist debate in the philosophy of set theory, this workshop will have a master class on potentialism, a series of several speakers, and a panel discussion. To be held 21-22 September 2022 at the University of Konstanz, Germany. (Contact organizers for Zoom access.)

I shall make several contributions to the meeting.

Master class tutorial on potentialism

Slides-Potentialism-Master-Class-Hamkins-Sep-2022

I shall give a master class tutorial on potentialism, an introduction to the general theory of potentialism that has been emerging in recent work, often developed as a part of research on set-theoretic pluralism, but just as often branching out to a broader application. Although the debate between potentialism and actualism in the philosophy of mathematics goes back to Aristotle, recent work divorces the potentialist idea from its connection with infinity and undertakes a more general analysis of possible mathematical universes of any kind. Any collection of mathematical structures forms a potentialist system when equipped with an accessibility relation (refining the submodel relation), and one can define the modal operators of possibility $\Diamond\varphi$, true at a world when $\varphi$ is true in some larger world, and necessity $\Box\varphi$, true in a world when $\varphi$ is true in all larger worlds. The project is to understand the structures more deeply by understanding their modal nature in the context of a potentialist system. The rise of modal model theory investigates very general instances of potentialist system, for sets, graphs, fields, and so on. Potentialism for the models of arithmetic often connects with deeply philosophical ideas on ultrafinitism. And the spectrum of potentialist systems for the models of set theory reveals fundamentally different conceptions of set-theoretic pluralism and possibility.

The multiverse view on the axiom of constructibility

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I shall give a talk on the multiverse perspective on the axiom of constructibility. Set theorists often look down upon the axiom of constructibility V=L as limiting, in light of the fact that all the stronger large cardinals are inconsistent with this axiom, and furthermore the axiom expresses a minimizing property, since $L$ is the smallest model of ZFC with its ordinals. Such views, I argue, stem from a conception of the ordinals as absolutely completed. A potentialist conception of the set-theoretic universe reveals a sense in which every set-theoretic universe might be extended (in part upward) to a model of V=L. In light of such a perspective, the limiting nature of the axiom of constructibility tends to fall away.

Panel discussion: The multiverse view—challenges for the next ten years

This will be a panel discussion on the set-theoretic multiverse, with panelists including myself, Carolin Antos-Kuby, Giorgio Venturi, and perhaps others.

Pointwise definable end-extensions of the universe, Sophia 2022, Salzburg

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This will be an online talk for the Salzburg Conference for Young Analytical Philosophy, the SOPhiA 2022 Salzburgiense Concilium Omnibus Philosophis Analyticis, with a special workshop session Reflecting on ten years of the set-theoretic multiverse. The workshop will meet Thursday 8 September 2022 4:00pm – 7:30pm.

The name of the workshop (“Reflecting on ten years…”), I was amazed to learn, refers to the period since my 2012 paper, The set-theoretic multiverse, in the Review of Symbolic Logic, in which I had first introduced my arguments and views concerning set-theoretic pluralism. I am deeply honored by this workshop highlighting my work in this way and focussing on the developments growing out of it.

In this talk, I shall engage in that discussion by presenting some very new work connecting several topics that have been prominent in discussions of the set-theoretic multiverse, namely, set-theoretic potentialism and pointwise definability.

Abstract. Using the universal algorithm and its generalizations, I shall present new work on the possibility of end-extending any given countable model of arithmetic or set theory to a pointwise definable model, one in which every object is definable without parameters. Every countable model of Peano arithmetic, for example, admits an end-extension to a pointwise definable model. And similarly, every countable model of ZF set theory admits an end-extension to a pointwise definable model of ZFC+V=L, as well as to pointwise definable models of other sufficient theories, accommodating large cardinals. I shall discuss the philosophical significance of these results in the philosophy of set theory with a view to potentialism and the set-theoretic multiverse.

Slides-Hamkins-Pointwise-definable-end-extensions-SOPhiA-2022

Fregean abstraction in set theory—a deflationary account, Italian Philosophy of Mathematics, September 2022

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This will be a talk for the conference Philosophy of Mathematics: Foundations, Definitions and Axioms, the Fourth International Conference of the Italian Network for the Philosophy of Mathematics, 29 September to 1 October 2022.

Abstract. The standard set-theoretic distinction between sets and classes instantiates in important respects the Fregean distinction between objects and concepts, for in set theory we commonly take the universe of sets as a realm of objects to be considered under the guise of diverse concepts, the definable classes, each serving as a predicate on that domain of individuals. Although it is commonly held that in a very general manner, there can be no association of classes with objects in a way that fulfills Frege’s Basic Law V, nevertheless, in the ZF framework, it turns out that we can provide a completely deflationary account of this and other Fregean abstraction principles. Namely, there is a mapping of classes to objects, definable in set theory in senses I shall explain (hence deflationary), associating every first-order parametrically definable class $F$ with a set object $\varepsilon F$, in such a way that Basic Law V is fulfilled: $$\varepsilon F =\varepsilon G\iff\forall x\ (Fx\leftrightarrow Gx).$$ Russell’s elementary refutation of the general comprehension axiom, therefore, is improperly described as a refutation of Basic Law V itself, but rather refutes Basic Law V only when augmented with powerful class comprehension principles going strictly beyond ZF. The main result leads also to a proof of Tarski’s theorem on the nondefinability of truth as a corollary to Russell’s argument. A central goal of the project is to highlight the issue of definability and deflationism for the extension assignment problem at the core of Fregean abstraction.

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Nonlinearity and illfoundedness in the hierarchy of consistency strength and the question of naturality, Italy (AILA), September 2022

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This will be a talk for the meeting of The Italian Association for Logic and its Applications (AILA) in Caserta, Italy 12-15 September 2022.

Abstract. Set theorists and philosophers of mathematics often point to a mystery in the foundations of mathematics, namely, that our best and strongest mathematical theories seem to be linearly ordered and indeed well-ordered by consistency strength. Why should it be? The phenomenon is thought to carry profound significance for the philosophy of mathematics, perhaps pointing us toward the ultimately correct mathematical theories, the “one road upward.” 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, however, are often dismissed as unnatural or as Gödelian trickery. In this talk, I aim to rebut that criticism 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.

AILA-Caserta-2022-Slides-Hamkins-Nonlinearity-and-illfoundedness-in-the-hierarchy-of-consistency-strength-and-the-question-of-naturalityDownload

Set theory inside out: realizing every inner model theory in an end extension, European Set Theory Conference, September 2022

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This will be a talk for the European Set Theory Conference 2022 in Turin, Italy 29 August – 2 September 2022.

Abstract. Every countable model of ZFC set theory with an inner model satisfying a sufficient theory must also have an end-extension satisfying that theory. For example, every countable model with a measurable cardinal has an end-extension to a model of $V=L[\mu]$; every model with extender-based large cardinals has an end-extension to a model of $V=L[\vec E]$; every model with infinitely many Woodin cardinals and a measurable above has an end-extension to a model of $\text{ZF}+\text{DC}+V=L(\mathbb{R})+\text{AD}$. These results generalize the famous Barwise extension theorem, of course, asserting that every countable model of ZF set theory admits an end-extension to a model of $\text{ZFC}+{V=L}$, a theorem which was simultaneously a technical culmination of Barwise’s pioneering methods 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 Barwise theorem that omits any need for infinitary logic and relies instead only on classical methods of descriptive set theory, while also providing the generalization I mentioned. This proof furthermore 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, a result holding important consequences for the nature of set-theoretic potentialism.  This work is joint with Kameryn J. Williams.

[bibtex key=”HamkinsWilliams2021:The-universal-finite-sequence”]