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Joel David Hamkins

mathematics and philosophy of the infinite

Joel David Hamkins

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Tag Archives: The Church of Logic

The Church of Logic podcast, April 2025

Posted on April 21, 2025 by Joel David Hamkins
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I was interviewed by Cody Roux for The Church of Logic podcast—a fascinating sweeping conversation on issues in the philosophy of mathematics and set theory, including what I described as a fundamental dichotomy between two perspectives on the nature of mathematics and what it is all about. Cody and I have affinities with opposite sides of this dichotomy, which made for a fruitful exchange.

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Posted in Talks, Videos | Tagged Cody Roux, philosophy of logic, philosophy of mathematics, philosophy of set theory, podcast, The Church of Logic | Leave a reply

Infinitely More

Proof and the Art of Mathematics, MIT Press, 2020

Recent Comments

  • David Roberts on Skolem’s paradox and the countable transitive submodel theorem, Leeds Set Theory Seminar, May 2025
  • David Roberts on A potentialist conception of ultrafinitism, Columbia University, April 2025
  • Joel David Hamkins on A potentialist conception of ultrafinitism, Columbia University, April 2025
  • David Roberts on A potentialist conception of ultrafinitism, Columbia University, April 2025
  • David Roberts on A potentialist conception of ultrafinitism, Columbia University, April 2025

JDH on Twitter

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RSS Mathoverflow activity

  • Comment by Joel David Hamkins on Is existence of one step downshifting embeddings consistent with Stratified ZF?
    But also, this notion of embedding is very weak. Indeed, ZFC proves that there are embeddings $j:V\to V$ that are definable and not the identity. Being definable, they can appear in the replacement and separation axioms. I prove this in my paper: worldscientific.com/doi/abs/10.1142/S0219061313500062. But those $j$ will not have $j(\alpha)+1=\alpha$, and indeed, they don't even […]
  • Comment by Joel David Hamkins on Is existence of one step downshifting embeddings consistent with Stratified ZF?
    The injectivity requirement on $j$ follows from the embedding part by extensionality, since if $x\neq y$, then there is some $z$ in one of them and not the other, and so $j(z)$ will distinguish $j(x)$ and $j(y)$.
  • Comment by Joel David Hamkins on The club filter in definable preorders
    Well, that isn't really correct, since $\omega_1+1$ embeds into those $\omega_n$. But there is a version of the question, I suppose, that drops that hypothesis, and this is still interesting. Under AD, Jackson has investigated which cardinals are measurable, but I am unsure whether we know the cofinalities of the cardinals. An affirmative answer to […]
  • Comment by Joel David Hamkins on The club filter in definable preorders
    Under AD, we know $\omega_1$ and $\omega_2$ are measurable, via the club filter, and $\omega_n$ is not measurable $3\leq n
  • Comment by Joel David Hamkins on Is there a general way to translate well-founded and non-well founded models of theories?
    It is the same with Boffa, which has many automorphisms, although one needs parameters to define the automorphisms.
  • Comment by Joel David Hamkins on Is there a general way to translate well-founded and non-well founded models of theories?
    The whole universe. For example, if we add the axiom that there are exactly two Quine atoms, and everything else is generated from them in a well-founded hierarchy, then swapping them is a definable automorphism of the universe.
  • Comment by Joel David Hamkins on Is there a general way to translate well-founded and non-well founded models of theories?
    That won't be true, since ZFC-Reg+exists ill-fdd has extensions with definable automorphisms, and these can never be bi-interpretable with ZFC, which is definably rigid.
  • Comment by Joel David Hamkins on Is there a general way to translate well-founded and non-well founded models of theories?
    Yes, that is what I meant. For example, Aczel's anti-foundational theory (with choice) is bi-interpretable with ZFC.

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