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.

J. D. Hamkins, “Every countable model of arithmetic or set theory has a pointwise definable end extension,” mathematics arXiv, 2022. [Bibtex]

@ARTICLE{Hamkins:Every-countable-model-of-arithmetic-or-set-theory-has-a-pointwise-definable-end-extension,
author = {Joel David Hamkins},
title = {Every countable model of arithmetic or set theory has a pointwise definable end extension},
journal = {mathematics arXiv},
year = {2022},
volume = {},
number = {},
pages = {},
month = {},
note = {},
abstract = {},
keywords = {},
source = {},
doi = {10.48550/ARXIV.2209.12578},
eprint = {2209.12578},
archivePrefix={arXiv},
primaryClass={math.LO},
url = {http://jdh.hamkins.org/pointwise-definable-end-extensions},
}

Abstract. According to the math tea argument, there must be real numbers that we cannot describe or define, because there are uncountably many real numbers, but only countably many definitions. And yet, the existence of pointwise definable models of set theory, in which every individual is definable without parameters, challenges this conclusion. In this article, I introduce a flexible new method for constructing pointwise definable models of arithmetic and set theory, showing furthermore that every countable model of Zermelo-Fraenkel ZF set theory and of Peano arithmetic PA has a pointwise-definable end extension. In the arithmetic case, I use the universal algorithm and its $\Sigma_n$ generalizations to build a progressively elementary tower making any desired individual $a_n$ definable at each stage $n$, while preserving these definitions through to the limit model, which can thus be arranged to be pointwise definable. A similar method works in set theory, and one can moreover achieve $V=L$ in the extension or indeed any other suitable theory holding in an inner model of the original model, thereby fulfilling the resurrection phenomenon. For example, every countable model of ZF with an inner model with a measurable cardinal has an end extension to a pointwise-definable model of $\text{ZFC}+V=L[\mu]$.

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.

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

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.

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.

Abstract. Many set theorists point to the linearity phenomenon in the hierarchy of consistency strength, by which natural theories tend to be linearly ordered and indeed well ordered by consistency strength. Why should it be linear? In this paper I present counterexamples, natural instances of nonlinearity and illfoundedness in the hierarchy of large cardinal consistency strength, as natural or as nearly natural as I can make them. I present diverse cautious enumerations of ZFC and large cardinal set theories, which exhibit incomparability and illfoundedness in consistency strength, and yet, I argue, are natural. I consider the philosophical role played by “natural” in the linearity phenomenon, arguing ultimately that we should abandon empty naturality talk and aim instead to make precise the mathematical and logical features we had found desirable.

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.

This is currently a draft version only of my article-in-progress on the topic of linearity in the hierarchy of consistency strength, especially with large cardinals. Comments are very welcome, since I am still writing the article. Please kindly send me comments by email or just post here.

This article will be the basis of the Weeks 7 & 8 discussion in the Graduate Philosophy of Logic seminar I am currently running with Volker Halbach at Oxford in Hilary term 2021.

I present instances of nonlinearity and illfoundedness in the hierarchy of large cardinal consistency strength—as natural or as nearly natural as I can make them—and consider philosophical aspects of the question of naturality with regard to this phenomenon.

It is a mystery often mentioned in the foundations of mathematics, a fundamental phenomenon to be explained, that our best and strongest mathematical theories seem to be linearly ordered and indeed well-ordered by consistency strength. Given any two of the familiar large cardinal hypotheses, for example, generally one of them will prove the consistency of the other.

Why should it be linear? Why should the large cardinal notions line up like this, when they often arise from completely different mathematical matters? Measurable cardinals arise from set-theoretic issues in measure theory; Ramsey cardinals generalize ideas in graph coloring combinatorics; compact cardinals arise with compactness properties of infinitary logic. Why should these disparate considerations lead to principles that are linearly related by direct implication and consistency strength?

The phenomenon is viewed by many in the philosophy of mathematics as significant in our quest for mathematical truth. In light of Gödel incompleteness, after all, we must eternally seek to strengthen even our best and strongest theories. Is the linear hierarchy of consistency strength directing us along the elusive path, the “one road upward” as John Steel describes it, toward the final, ultimate mathematical truth? That is the tantalizing possibility.

Meanwhile, we do know as a purely formal matter that the hierarchy of consistency strength is not actually well-ordered—it is ill-founded, densely ordered, and nonlinear. The statements usually used to illustrate these features, however, are weird self-referential assertions constructed in the Gödelian manner via the fixed-point lemma—logic-game trickery, often dismissed as unnatural.

Many set theorists claim that amongst the natural assertions, consistency strengths remain linearly ordered and indeed well ordered. H. Friedman refers to “the apparent comparability of naturally occurring logical strengths as one of the great mysteries of [the foundations of mathematics].” Andrés Caicedo says,

It is a remarkable empirical phenomenon that we indeed have comparability for natural theories. We expect this to always be the case, and a significant amount of work in inner model theory is guided by this belief.

Stephen G. Simpson writes:

It is striking that a great many foundational theories are linearly ordered by <. Of course it is possible to construct pairs of artificial theories which are incomparable under <. However, this is not the case for the “natural” or non-artificial theories which are usually regarded as significant in the foundations of mathematics. The problem of explaining this observed regularity is a challenge for future foundational research.

John Steel writes “The large cardinal hypotheses [the ones we know] are themselves wellordered by consistency strength,” and he formulates what he calls the “vague conjecture” asserting that

If T is a natural extension of ZFC, then there is an extension H axiomatized by large cardinal hypotheses such that T ≡ Con H. Moreover, ≤ Con is a prewellorder of the natural extensions of ZFC. In particular, if T and U are natural extensions of ZFC, then either T ≤ Con U or U ≤ Con T.

Peter Koellner writes

Remarkably, it turns out that when one restricts to those theories that “arise in nature” the interpretability ordering is quite simple: There are no descending chains and there are no incomparable elements—the interpretability ordering on theories that “arise in nature” is a wellordering.

Let me refer to this position as the natural linearity position, the assertion that all natural assertions of mathematics are linearly ordered by consistency strength. The strong form of the position, asserted by some of those whom I have cited above, asserts that the natural assertions of mathematics are indeed well-ordered by consistency strength. By all accounts, this view appears to be widely held in large cardinal set theory and the philosophy of set theory.

Despite the popularity of this position, I should like in this article to explore the contrary view and directly to challenge the natural linearity position.

Main Question. Can we find natural instances of nonlinearity and illfoundedness in the hierarchy of consistency strength?

I shall try my best.

You have to download the article to see my candidates for natural instances of nonlinearity in the hierarchy of large cardinal consistency strength, but I can tease you a little by mentioning that there are various cautious enumerations of the ZFC axioms which actually succeed in enumerating all the ZFC axioms, but with a strictly weaker consistency strength than the usual (incautious) enumeration. And similarly there are various cautious versions of the large cardinal hypothesis, which are natural, but also incomparable in consistency strength.

(Please note that it was Uri Andrews, rather than Uri Abraham, who settled question 16 with the result of theorem 17. I have corrected this from an earlier draft.)

This bried unpublished note (11 pages) contains an overview of the Gödel fixed-point lemma, along with several generalizations and applications, written for use in the Week 3 lecture of the Graduate Philosophy of Logic seminar that I co-taught with Volker Halbach at Oxford in Hilary term 2021. The theme of the seminar was self-reference, truth, and consistency strengths, and in this lecture we discussed the nature of Gödel’s fixed-point lemma and generalizations, with various applications in logic.

Gödel’s fixed-point lemma An application to the Gödel incompleteness theorem

Finite self-referential schemes An application to nonindependent disjunctions of independent sentences

Gödel-Carnap fixed point lemma Deriving the double fixed-point lemma as a consequence An application to the provability version of Yablo’s paradox

Kleene recursion theorem An application involving computable numbers An application involving the universal algorithm An application to Quine programs and Ouroborous chains

The talk will be held online via Zoom ID: 998 6013 7362.

Abstract. It is a mystery often mentioned in the foundations of mathematics that our best and strongest mathematical theories seem to be linearly ordered and indeed well-ordered by consistency strength. Given any two of the familiar large cardinal hypotheses, for example, generally one of them proves the consistency of the other. Why should this be? The phenomenon is seen as significant for the philosophy of mathematics, perhaps pointing us toward the ultimately correct mathematical theories. 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 are often dismissed as unnatural or as Gödelian trickery. In this talk, I aim to overcome that criticism—as well as I am able to—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.

The talk should be generally accessible to university logic students, requiring little beyond familiarity with the incompleteness theorem and some elementary ideas from computability theory.

Abstract. Recent years have seen a flurry of mathematical activity in set-theoretic and arithmetic potentialism, in which we investigate a collection of models under various natural extension concepts. These potentialist systems enable a modal perspective—a statement is possible in a model, if it is true in some extension, and necessary, if it is true in all extensions. We consider the models of ZFC set theory, for example, with respect to submodel extensions, rank-extensions, forcing extensions and others, and these various extension concepts exhibit different modal validities. In this talk, I shall describe the state of current developments, including the most recent tools and results.

This will be a talk for the Barcelona Set Theory Seminar, 28 October 2020 4 pm CET (3 pm UK). Contact Joan Bagaria bagaria@ub.edu for the access link.

Abstract. The Barwise extension theorem, asserting that every countable model of ZF set theory admits an end-extension to a model of ZFC+V=L, is both a technical culmination of the pioneering methods of Barwise 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 theorem that omits any need for infinitary logic and relies instead only on classical methods of descriptive set theory. This proof 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. The result has strong consequences for the nature of set-theoretic potentialism. This work is joint with Kameryn J. Williams.

J. D. Hamkins and K. J. Williams, “The $\Sigma_1$-definable universal finite sequence,” Journal of Symbolic Logic, 2021. [Bibtex]

@ARTICLE{HamkinsWilliams2021:The-universal-finite-sequence,
author = {Joel David Hamkins and Kameryn J. Williams},
title = {The $\Sigma_1$-definable universal finite sequence},
journal = {Journal of Symbolic Logic},
year = {2021},
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keywords = {},
eprint = {1909.09100},
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}

Abstract. We introduce the $\Sigma_1$-definable universal finite sequence and prove that it exhibits the universal extension property amongst the countable models of set theory under end-extension. That is, (i) the sequence is $\Sigma_1$-definable and provably finite; (ii) the sequence is empty in transitive models; and (iii) if $M$ is a countable model of set theory in which the sequence is $s$ and $t$ is any finite extension of $s$ in this model, then there is an end extension of $M$ to a model in which the sequence is $t$. Our proof method grows out of a new infinitary-logic-free proof of the Barwise extension theorem, by which any countable model of set theory is end-extended to a model of $V=L$ or indeed any theory true in a suitable submodel of the original model. The main theorem settles the modal logic of end-extensional potentialism, showing that the potentialist validities of the models of set theory under end-extensions are exactly the assertions of S4. Finally, we introduce the end-extensional maximality principle, which asserts that every possibly necessary sentence is already true, and show that every countable model extends to a model satisfying it.

This will be a talk for the Theory Seminar for the theory research group in Theoretical Computer Science at Queen Mary University of London. The talk will be held 4 June 2019 1:00 pm, ITL first floor.

Abstract. Curious, often paradoxical instances of self-reference inhabit deep parts of computability theory, from the intriguing Quine programs and Ouroboros programs to more profound features of the Gödel phenomenon. In this talk, I shall give an elementary account of the universal algorithm, showing how the capacity for self-reference in arithmetic gives rise to a Turing machine program $e$, which provably enumerates a finite set of numbers, but which can in principle enumerate any finite set of numbers, when it is run in a suitable model of arithmetic. In this sense, every function becomes computable, computed all by the same universal program, if only it is run in the right world. Furthermore, the universal algorithm can successively enumerate any desired extension of the sequence, when run in a suitable top-extension of the universe. An analogous result holds in set theory, where Woodin and I have provided a universal locally definable finite set, which can in principle be any finite set, in the right universe, and which can furthermore be successively extended to become any desired finite superset of that set in a suitable top-extension of that universe.

This will be a talk at the Institute of Logic, Language and Computation (ILLC) at the University of Amsterdam for events May 11-12, 2019. See Joel David Hamkins in Amsterdam 2019.

Abstract: Potentialism can be seen as a fundamentally model-theoretic notion, in play for any class of mathematical structures with an extension concept, a notion of substructure by which one model extends to another. Every such model-theoretic context can be seen as a potentialist framework, a Kripke model whose modal validities one can investigate. In this talk, I’ll explain the tools we have for analyzing the potentialist validities of such a system, with examples drawn from the models of arithmetic and set theory, using the universal algorithm and the universal definition.