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Joel David Hamkins

mathematics and philosophy of the infinite

Joel David Hamkins

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Tag Archives: Cody Roux

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

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Proof and the Art of Mathematics, MIT Press, 2020

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RSS Mathoverflow activity

  • Comment by Joel David Hamkins on Formal justification for Freiling's axiom of symmetry
    The title of your question doesn't seem to match the question(s) that you actually ask.
  • Comment by Joel David Hamkins on How to understand non-standard halting times?
    Just because PA is actually sound it doesn't follow that every model of PA thinks that PA is sound, and indeed, I have explained why some don't.
  • Comment by Joel David Hamkins on How to understand non-standard halting times?
    Well, in my example M thinks PA is inconsistent, so it thinks PA can prove anything. It doesn't agree that PA is sound, even though it happens to think every standard axiom of PA is true. It has problematic nonstandard axioms of PA.
  • Comment by Joel David Hamkins on How to understand non-standard halting times?
    Yes, that is totally right. The operation at finite stages is absolute.
  • Comment by Joel David Hamkins on How to understand non-standard halting times?
    Having a code of a proof that $p$ halts does not generally imply that one has a code of the halting computation. For example, in a model of PA+¬Con(PA), there will be codes of proofs that every single program halts, even though many of them do not, even in that model.
  • Answer by Joel David Hamkins for How to understand non-standard halting times?
    One of the key ideas underlying so much of our understanding of models of arithmetic, including the incompleteness theorems and so much more, is the arithmetization phenomenon. Arithmetization shows that essentially any finite combinatorial concept can be expressed using only the very basic language of arithmetic, with addition and multiplication only. One commonly takes the […]
  • Comment by Joel David Hamkins on Finding the largest integer describable with a string of symbols of predefined length
    @JackEdwardTisdell Yes, that is what I meant.
  • Comment by Joel David Hamkins on Does anyone still seriously doubt the consistency of $ZFC$?
    Well, I had said the conflation is present even in PA, which is of course much weaker than ACA. I don't see what "real-world" has to do with anything, since we have even less understanding of the physical world, which becomes more deeply mysterious the more we know about it, than of arithmetic. I am […]

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