Joel David Hamkins, “How the continuum hypothesis could have been a fundamental axiom,” Journal for the Philosophy of Mathematics (2024), arxiv:2407.02463.

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

The JPM will launch in September 2024. Meanwhile, the preprint pdf is available at arxiv.org/pdf/2407.02463.

See also this talk I gave on the topic at the University of Oslo:

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

Abstract. We discuss the accuracy of the attribution commonly given to Turing (1936) for the computable undecidability of the halting problem, eventually coming to a nuanced conclusion.

The halting problem is the decision problem of determining whether a given computer program halts on a given input, a problem famously known to be computably undecidable. In the computability theory literature, one quite commonly finds attribution for this result given to Alan Turing (1936), and we should like to consider the extent to which these attributions are accurate. After all, the term halting problem, the modern formulation of the problem, as well as the common self-referential proof of its undecidability, are all—strictly speaking—absent from Turing’s work. However, Turing does introduce the concept of an undecidable decision problem, proving that what he calls the circle-free problem is undecidable and subsequently also that what we call the symbol-printing problem, to decide if a given program will ever print a given symbol, is undecidable. This latter problem is easily seen to be computably equivalent to the halting problem and can arguably serve in diverse contexts and applications in place of the halting problem—they are easily translated to one another. Furthermore, Turing laid down an extensive framework of ideas sufficient for the contemporary analysis of the halting problem, including: the definition of Turing machines; the labeling of programs by numbers in a way that enables programs to be enumerated and also for them to be given as input to other programs; the existence of a universal computer; the undecidability of several problems that, like the halting problem, take other programs as input, including the circle-free problem, the symbol-printing problem, and the infinite-symbol-printing problem, as well as the Hilbert-Ackermann Entscheidungsproblem. In light of these facts, and considering some general cultural observations, by which mathematical attributions are often made not strictly for the exact content of original work, but also generously in many cases for the further aggregative insights to which those ideas directly gave rise, ultimately we do not find it unreasonable to offer qualified attribution to Turing for the undecidability of the halting problem. That said, we also find it incorrect to suggest that one will find a discussion of the halting problem or a proof of its undecidability in Turing (1936).

Read the full paper at the mathematics arxiv:2407.00680. (pdf download available)

This will be a talk for the Axe Histoire et Philosophie des mathématiques, Séminaire PhilMath Intersem 2023, a collaborative event sponsored by the University of Notre Dame and le laboratoire SPHERE, Paris. The Intersem runs several weeks, but my talk will be 9 June.

Abstract. The 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 εF, in such a way that Basic Law V is fulfilled:

εF=εG ⇔ ∀x (Fx ⇔ 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, one amounting, I argue, to a truth predicate in Frege’s system. The main result therefore leads to a proof of Tarski’s theorem on the nondefinability of truth as a corollary to Russell’s argument, independently of Gödel. 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.

Abstract. I shall speak on the surprising strength of the second-order reflection principle in the context of set theory with abundant urelements. The theory GBcU with the abundant urelement axiom and second-order reflection is bi-interpretable with a strengthening of KM with a supercompact cardinal. This is joint work with Bokai Yao.

Abstract. Every mathematical structure has an elementary extension to a pseudo-countable structure, one that is seen as countable inside a suitable class model of set theory, even though it may actually be uncountable. This observation, proved easily with the Boolean ultrapower theorem, enables a sweeping generalization of results concerning countable models to a rich realm of uncountable models. The Barwise extension theorem, for example, holds amongst the pseudo-countable models—every pseudo-countable model of ZF admits an end extension to a model of ZFC+V=L. Indeed, the class of pseudo-countable models is a rich multiverse of set-theoretic worlds, containing elementary extensions of any given model of set theory and closed under forcing extensions and interpreted models, while simultaneously fulfilling the Barwise extension theorem, the Keisler-Morley theorem, the resurrection theorem, and the universal finite sequence theorem, among others.

A playful account of symmetry, contributed as a chapter to a larger work, The Language of Symmetry, edited by Benedict Rattigan, Denis Noble, and Afiq Hatta, a collection of essays on symmetry that were also the basis of an event at the British Museum, The Language of Symmetry.

Abstract. Let me tell a mathematician’s tale about symmetry. We begin with playful curiosity about a concrete elementary case—the symmetries of the letters of the alphabet, for instance. Seeking the essence of symmetry, however, we are pushed toward abstraction, to other shapes and higher dimensions. Beyond the geometric figures, we consider the symmetries of an arbitrary mathematical structure—why not the symmetries of the symmetries? And then, of course, we shall have the symmetries of the symmetries of the symmetries, and so on, iterating transfinitely. Amazingly, this process culminates in a sublime self-similar group of symmetries that is its own symmetry group, a self-similar self-similarity.

Download my essay for more…or order the book for the complete set!

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 $\text{ZFC}+V=L[\mu]$.

Abstract. The standard treatment of sets and definable classes in first-order Zermelo-Fraenkel set theory accords in many respects with the Fregean foundational framework, such as the distinction between objects and concepts. Nevertheless, in set theory we may define an explicit association of definable classes with set objects $F\mapsto\varepsilon F$ in such a way, I shall prove, to realize Frege’s Basic Law V as a ZF theorem scheme, Russell notwithstanding. A similar analysis applies to the Cantor-Hume principle and to Fregean abstraction generally. Because these extension and abstraction operators are definable, they provide a deflationary account of Fregean abstraction, one expressible in and reducible to set theory—every assertion in the language of set theory allowing the extension and abstraction operators $\varepsilon F$, $\# G$, $\alpha H$ is equivalent to an assertion not using them. The analysis thus sidesteps Russell’s argument, which is revealed not as a refutation of Basic Law V as such, but rather as a version of Tarski’s theorem on the nondefinability of truth, showing that the proto-truth-predicate “$x$ falls under the concept of which $y$ is the extension” is not expressible.

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. After reviewing various natural bi-interpretations in urelement set theory, including second-order set theories with urelements, we explore the strength of second-order reflection in these contexts. Ultimately, we prove, second-order reflection with the abundant atom axiom is bi-interpretable and hence also equiconsistent with the existence of a supercompact cardinal. The proof relies on a reflection characterization of supercompactness, namely, a cardinal $\kappa$ is supercompact if and only if every $\Pi^1_1$ sentence true in a structure $M$ (of any size) containing $\kappa$ in a language of size less than $\kappa$ is also true in a substructure $m\prec M$ of size less than $\kappa$ with $m\cap\kappa\in\kappa$.

Abstract. I consider the natural infinitary variations of the games Wordle and Mastermind, as well as their game-theoretic variations Absurdle and Madstermind, considering these games with infinitely long words and infinite color sequences and allowing transfinite game play. For each game, a secret codeword is hidden, which the codebreaker attempts to discover by making a series of guesses and receiving feedback as to their accuracy. In Wordle with words of any size from a finite alphabet of $n$ letters, including infinite words or even uncountable words, the codebreaker can nevertheless always win in $n$ steps. Meanwhile, the mastermind number 𝕞, defined as the smallest winning set of guesses in infinite Mastermind for sequences of length $\omega$ over a countable set of colors without duplication, is uncountable, but the exact value turns out to be independent of ZFC, for it is provably equal to the eventually different number $\frak{d}({\neq^*})$, which is the same as the covering number of the meager ideal $\text{cov}(\mathcal{M})$. I thus place all the various mastermind numbers, defined for the natural variations of the game, into the hierarchy of cardinal characteristics of the continuum.

Abstract. We introduce the game of infinite Hex, extending the familiar finite game to natural play on the infinite hexagonal lattice. Whereas the finite game is a win for the first player, we prove in contrast that infinite Hex is a draw—both players have drawing strategies. Meanwhile, the transfinite game-value phenomenon, now abundantly exhibited in infinite chess and infinite draughts, regrettably does not arise in infinite Hex; only finite game values occur. Indeed, every game-valued position in infinite Hex is intrinsically local, meaning that winning play depends only on a fixed finite region of the board. This latter fact is proved under very general hypotheses, establishing the conclusion for all simple stone-placing games.

A joint paper with Davide Leonessi, in which we prove that every countable ordinal arises as the game value of a position in infinite draughts, and this result is optimal for games having countably many options at each move. In short, the omega one of infinite draughts is true omega one.

Abstract. Infinite draughts, or checkers, is played just like the finite game, but on an infinite checkerboard extending without bound in all four directions. We prove that every countable ordinal arises as the game value of a position in infinite draughts. Thus, there are positions from which Red has a winning strategy enabling her to win always in finitely many moves, but the length of play can be completely controlled by Black in a manner as though counting down from a given countable ordinal.

Abstract. We show that there is an arithmetical formula $\varphi$ such that ZF proves that $\varphi$ is independent of PA and yet, unlike other arithmetical independent statements, the truth value of $\varphi$ cannot at present be established in ZF or in any other trusted metatheory. In fact we can choose an example of such a formula $\varphi$ such that ZF proves that $\varphi$ is equivalent to the twin prime conjecture. We conclude with a discussion of notion of trustworthy theory and a sharper version of the result.

In this insightful and remarkable work, Professor Novaes defends and explores at length the philosophical thesis that mathematical proof and deduction generally has a fundamentally dialogical nature, proceeding in a back-and-forth dialogue between two semi-adversarial but collaborative actors, the Prover and the Skeptic, who together aim to find mathematical insight. This view of proof-as-dialogue, she argues, carries explanatory power for the philosophy of mathematical practice, explaining diverse aspects of proof-writing, refereeing, and more, including the multifaceted roles of proof, including proof as verification, proof as certification, proof for communication and persuasion, proof as explanation, and proof as a driver of mathematical innovation.

In extensive, refined scholarly work, Novaes explores the historical and intellectual roots of the dialogical perspective on deduction, tracing the idea from ancient times through medieval philosophy and into the present day, including case studies of current mathematical developments, such as Mochizuki’s claimed proof of the abc conjecture, as well as recent psychological experiments on the role of group reasoning in resolving certain well-known disappointing failures of rationality, such as in the Wason card experiment. Truly fascinating.

On the basis of her work, I have nominated Novaes for the Lakatos Award (given annually “for an outstanding contribution to the philosophy of science, widely interpreted, in the form of a book published in English during the current year or the previous five years”). Lakatos himself, of course, was a friend of dialogical mathematics—his famous Proofs and Refutations proceeds after all in a dialogue between mathematicians of different philosophical outlooks. Novaes engages with Lakatos’s work explicitly, pointing out the obvious parallels, but also highlighting important differences between her Prover/Skeptic dialogical account and the kind of proof dialogues appearing in Proofs and Refutations. In light of this connection with Lakatos, I would find it especially fitting for Novaes to win the Lakatos Award.