Tag Archives: potentialism

Potentialist conceptions of infinity, Peking University, June 2025

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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.

Slides – Potentialist conceptions of infinity – Hamkins – Peking University June 2025Download

Lectures on Set Theory, Beijing, June 2025

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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.

A potentialist conception of ultrafinitism, Columbia University, April 2025

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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.

Slides – Ultrafinitism – Columbia 2025 – HamkinsDownload

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

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 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.

What is second-order predicate modal logic? FoMoLo Seminar, February 2024

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This will be a talk for the First-order Modal Logic (FoMoLo) Seminar, 12 February 2024. The talk will take place online via Zoom—contact the organizers for access.

Abstract. What is or should be the potentialist account of classes? There are several natural implementations of second-order logic in a modal potentialist setting, which arise from differing philosophical conceptions of the nature of the second-order resources. I shall introduce the proposals, analyze their comparative expressive and interpretative powers, and explain how various philosophical attitudes are fulfilled or not for each proposal. This is joint work in progress with Øystein Linnebo. 

What is potentialist second-order logic? Konstanz Actualism and Potentialism Conference 2023

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This is a talk for the Actualism and Potentialism Conference at the University of Konstanz, 28-29 September 2023. Also on Zoom: 928 0804 3434.

Abstract. What is or should be the potentialist account of classes? It turns out that there are several natural implementations of second-order logic in a modal potentialist setting, which arise from differing philosophical conceptions of the nature of the second-order resources. I shall introduce the proposals, analyze their comparative expressive and interpretative powers, and explain how various philosophical attitudes are fulfilled or not for each proposal. This is joint work in progress with Øystein Linnebo. 

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

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 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.

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UW Madison Logic Seminar, Joel David Hamkins, April 4, 2023
Slides-Leibnizian-end-extensions-Madison-2023Download

Infinity, University of Notre Dame, Spring 2023

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Infinity

Philosophy 20607 01 (32582)

University of Notre Dame                                                                              Spring 2023

Instructor: Joel David Hamkins, O’Hara Professor of Philosophy and Mathematics
3:30-4:45 Tuesdays + Thursdays, DeBartolo Hall 208

Course Description. This course will be a mathematical and philosophical exploration of infinity, covering a wide selection of topics illustrating this rich, fascinating concept—the mathematics and philosophy of the infinite.

Along the way, we shall find paradox and fun—and all my favorite elementary logic conundrums and puzzles. It will be part of my intention to reveal what I can of the quirky side of mathematics and logic in its connection with infinity, but with a keen eye open for when issues happen to engage with philosophically deeper foundational matters.

The lectures will be based on the chapters of my forthcoming book, The Book of Infinity, currently in preparation, and currently being serialized and made available on the Substack website as I explain below.

Topics. Among the topics we shall aim to discuss will be:

  • The Book of Numbers
  • Zeno’s paradox
  • The infinite coastline paradox
  • Supertasks
  • Largest number contest
  • The googol plex chitty bang stack hierarchy
  • Galileo’s Salviati on infinity
  • Hilbert’s Grand Hotel
  • The uncountable
  • How to count (to infinity and beyond!)
  • Slaying the Hydra
  • Transfinite recursion
  • The continuum hypothesis
  • The axiom of choice
  • Orders of infinity
  • The lattice of subsets of ℕ
  • Potential versus actual infinity
  • Confounding puzzles of infinity
  • Infinite liars
  • Infinite utilitarianism
  • Infinite computation
  • Infinite games
  • Indescribable numbers
  • Extremely remote events of enormous consequence
  • The sand reckoner
  • Paradox in high dimension
  • The outer limits of reason
  • Puzzles of epistemic logic and the problem of common knowledge

Mathematical background. The course will at times involve topics and concepts of a fundamentally mathematical nature, but no particular mathematical background or training will be assumed. Nevertheless, it is expected that students be open to mathematical thinking and ideas, and furthermore it is a core aim of the course to help develop the student’s mastery over various mathematical concepts connected with infinity.  

Readings. The lectures will be based on readings from the topic list above that will be made available on my Substack web page, Infinitely More. Readings for the topic list above will be gradually released there during the semester. Each reading will consist of a chapter essay my book-in-progress, The Book of Infinity, which is being serialized on the Substack site specifically for this course. In some weeks, there will be supplemental readings from other sources.

Student access. I will issue subscription invitations to the Substack site for all registered ND students using their ND email, with free access to the site during the semester, so that students can freely access the readings.  Students are free to manage their subscriptions however they see fit. Please inform me of any access issues. There are some excellent free Substack apps available for Apple iOS and Android for reading Substack content on a phone or other device.

Discussion forum. Students are welcome to participate in the discussion forums provided with the readings to discuss the topics, the questions, to post answer ideas, or engage in the discussion there. I shall try to participate myself by posting comments or hints.

Homework essays. Students are expected to engage fully with every topic covered in the class. Every chapter concludes with several Questions for Further Thought, with which the students should engage. It will be expected that students complete approximately half of the Questions for Further thought. Each question that is answered should be answered essay-style with a mini-essay of about half a page or more.

Extended essays. A student may choose at any time to answer one of the Questions for Further Thought more fully with a more extended essay of two or three pages, and in this case, other questions on that particular topic need not be engaged. Every student should plan to exercise this option at least twice during the semester.

Final exam.  There will be a final exam consisting of questions similar to those in the Questions for Further Thought, covering every topic that was covered in the course. The final grade will be based on the final exam and on the submitted homework solutions.

Open Invitation. Students outside of Notre Dame are welcome to follow along with the Infinity course, readings, and online discussion. Simply subscribe at Infinitely More, keep up with the readings and participate in the discussions we shall be having in the forums there.

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 ◊φ, true at a world when φ is true in some larger world, and necessity ◻φ, true in a world when φ 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

Slides-Multiverse-view-on-VL-Hamkins-Konstanz-2022Download

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

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[μ]; every model with extender-based large cardinals has an end-extension to a model of V=L[E→]; every model with infinitely many Woodin cardinals and a measurable above has an end-extension to a model of ZF+DC+V=L(R)+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 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 Σ1-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”]

Potentialism and implicit actualism in the foundations of mathematics, Notre Dame, March 2021

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This will be a talk for the Department Colloquium of the Philosophy Department of the University of Notre Dame, 26 March 12 pm EST (4pm GMT).

Potentialist perspectives

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.