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Joel David Hamkins

mathematics and philosophy of the infinite

Joel David Hamkins

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Tag Archives: Rahul Sam

The Gödel incompleteness phenomenon, interview with Rahul Sam

Posted on January 14, 2024 by Joel David Hamkins
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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.

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Posted in Talks, Videos | Tagged large cardinals, multiverse, philosophy of mathematics, pluralism, potentialism, Rahul Sam | Leave a reply

Infinitely More

How we might have viewed the continuum hypothesis as a fundamental axiom necessary for mathematics

By mounting a philosophical historical thought experiment, I argue that our attitude toward the continuum hypothesis could easily have been very different than it is.

Joel David Hamkins
15 HR AGO
4
6
Take my Philosophy and Logic of Games final exam!

Can you pass the exam for my games course?

Joel David Hamkins
May 14
8
10
Pushpast

Can the triangles push past the circles?

Joel David Hamkins
May 7
1
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→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(α)+1=α, 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≠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 ω1+1 embeds into those ωn. 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 ω1 and ω2 are measurable, via the club filter, and ωn 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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