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Joel David Hamkins

mathematics and philosophy of the infinite

Joel David Hamkins

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Tag Archives: Mindscape

Puzzles of reality and infinity, Mindscape Podcast

Posted on July 15, 2024 by Joel David Hamkins
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I was interviewed by Sean Carroll for his Mindscape Podcast, broadcast 15 July 2024.

282 | Joel David Hamkins on Puzzles of Reality and Infinity

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Posted in Talks, Videos | Tagged Gödel, incompleteness, Mindscape, multiverse, philosophy of mathematics, Sean Carroll | Leave a reply

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Proof and the Art of Mathematics, MIT Press, 2020

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Recent Comments

  • Joel David Hamkins on Lectures on Set Theory, Beijing, June 2025
  • Joel David Hamkins on Lectures on Set Theory, Beijing, June 2025
  • Mohammad Golshani on Lectures on Set Theory, Beijing, June 2025
  • Jack Edward Tisdell on Lectures on Set Theory, Beijing, June 2025
  • Joel David Hamkins on Skolem’s paradox and the countable transitive submodel theorem, Leeds Set Theory Seminar, May 2025

JDH on Twitter

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RSS Mathoverflow activity

  • Answer by Joel David Hamkins for A question about Borel code
    The property of r being a Borel code is a Π11 property, since everything about it is easy arithmetic except asserting that the tree you are labeling is well-founded, and that brings it up to Π11. If r is indeed a Borel code, then to say that x∈Ar is complexity Δ11(r), since you can […]
  • Comment by Joel David Hamkins on A question about Borel code
    By a Borel map, you mean a Borel function from R to R? Do you insist on totality?
  • Comment by Joel David Hamkins on Strength of the incompleteness theorem as an axiom
    There are often quite interesting math discussions occurring on Twitter, bluesky, Reddit, YouTube, etc, and I think this is great. Sometimes they link to MO or each other, and sometimes MO links to them. In my view, we should link to the relevant math ideas wherever they may be found.
  • Answer by Joel David Hamkins for Strength of the incompleteness theorem as an axiom
    You mention in your title and first paragraph the idea of taking the incompleteness theorem "as an axiom". But the incompleteness theorem, taken as a sweeping statement about all sufficient theories of arithmetic, is a theorem—it is provable in PA and indeed in much weaker theories. Therefore if our axioms include PA or those weaker […]
  • Comment by Joel David Hamkins on What is the relationship between "translation" and time complexity?
    Thanks for this info.
  • Comment by Joel David Hamkins on References for incompleteness proofs using infinite trees or König's lemma
    Sorry, no. But you could send the link to this post, at least, for this argument. Perhaps there is another argument...If you find out, please post it as another answer.
  • Answer by Joel David Hamkins for Model of ZFC In Which Martin's Cone Theorem Fails?
    If you have a projective well-ordering of the reals in order type ω1, as you do in L, then you can simply build one by transfinite recursion. Well order the reals, which gives you also a well-ordering of the Turing degrees. There are continuum many cones, and so we can build our desired set in […]
  • Answer by Joel David Hamkins for Computably constructing a set that is not in a free ultrafilter
    The answer is negative. For some ultrafilters U and processes S:f↦S(f), there is no total computable increasing function f with S(f)∉U. Let me explain how to make this situation occur. What I claim is that we can produce a uniform computable process f↦S(f), defined in terms of the programs, such that for […]

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