Abstract. The principle of covering reflection holds of a cardinal κ if for every structure B in a countable first-order language there is a structure A of size less than κ, such that B is covered by elementary images of A in B. Is there any such cardinal? Is the principle consistent? Does it have large cardinal strength? This is joint work with myself, Nai-Chung Hou, Andreas Lietz, and Farmer Schlutzenberg.
I had an enjoyable little discussion with Joe Murray of The Human Podcast, part of his new series, called 10 questions in 10 minutes, in which he asks his interview subjects for short answers to ten quick questions on their topic. Here is our conversation:
Joe was adamant about the 1 minute timeline for each question, and was holding up timers and giving me the 5 second warning and so forth, but of course, it was simply impossible! There was no way for me to contain my answers to the time limit.
Meanwhile, you can follow through to our previous, longer discussion here:
I was interviewed by Francesco Cavina for the Back to the Stone Age series on May 17, 2024, with a sweeping discussion of the philosophy of set theory, infinity, the continuum hypothesis, beauty in mathematics, and much more.
I was interviewed by The Human Podcast on 17 May 2024. Please enjoy our sweeping conversation about nature of infinity, the nature of abstract mathematical existence, the applicability of mathematical abstractions to physical reality, and more. At the end, you will see that I am caught completely at a loss in answer to the question, “What is it to live a good life?”.
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.
We had a sweeping discussion touching upon many issues in the philosophy of mathematics, including the nature of mathematical truth, mathematical abstraction, the nature of mathematical existence, the meaning and role of proof in mathematics, the completeness theorem, the incompleteness phenomenon, infinity, and a discussion about the motivations that one might have for studying mathematics.
This will be an online Zoom talk for the Boston Computaton Club, a graduate seminar in computer science at Northeastern University, 16 June 12pm EST (note change in date/time). Contact the organizers for the Zoom link.
Abstract: Many familiar finite games admit natural infinitary analogues, which may captivate and challenge us with 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.
This will be a talk for the CUNY Set Theory Seminar on May 5, 2023 10am. Contact the organizers for the Zoom link.
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.
Abstract: I shall explore several senses in which set-theoretic forcing can be seen as a computational process on the models of set theory. Given an oracle for the atomic or elementary diagram of a model (M,∈M) of set theory, for example, there are senses in which one may compute M-generic filters G⊂ℙ∈M over that model and compute the diagrams of the corresponding forcing extensions M[G]. Meanwhile, no such computational process is functorial, for there must always be isomorphic alternative presentations of the same model of set theory that lead by the computational process to non-isomorphic forcing extensions. Indeed, there is no Borel function providing generic filters that is functorial in this sense. This is joint work with myself, Russell Miller and Kameryn Williams.
The talk took place in “The Barn” in the upper space between the Reyerson Laboratory and Eckhart Hall, where the University of Chicago Department of Mathematics is located:
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:
I was interviewed by Nathan Ormond for a discussion on Frege’s philosophy of mathematics for his YouTube channel, Digital Gnosis, on 10 December 2021 at 4pm.
The interview concludes with a public comment and question & answer session.
Abstract. The standard treatment of sets and classes in Zermelo-Fraenkel set theory instantiates in many respects the Fregean foundational distinction between objects and concepts, for in set theory we commonly take the sets as 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 often asserted that there can be no association of classes with objects in a way that fulfills Frege’s Basic Law V, nevertheless, in the ZF framework I have described, it turns out that Basic Law V does hold, and provably so, along with other various Fregean abstraction principles. These principles are consequences of Zermelo-Fraenkel ZF set theory in the context of all its definable classes. Namely, there is an injective mapping from classes to objects, definable in senses I shall explain, 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.
I would like to thank the many people who had made helpful suggestions concerning my poster, including Andrej Bauer and especially Saul Schleimer, who offered many detailed suggestions.
I was interviewed 26 August 2021 by mathematician Daniel Rubin on his show, and we had a lively, wideranging discussion spanning mathematics, infinity, and the philosophy of mathematics. Please enjoy!
