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Joel David Hamkins

mathematics and philosophy of the infinite

Joel David Hamkins

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Tag Archives: Francesco Cavina

Mathematics, Philosophy of Set Theory and Infinity, Back to the Stone Age interview, May 2024

Posted on June 7, 2024 by Joel David Hamkins
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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.

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Posted in Talks, Videos | Tagged continuum hypothesis, Francesco Cavina, infinity, philosophy of set theory | Leave a reply

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  • Joel David Hamkins on Infinite Sudoku and the Sudoku game
  • Joel David Hamkins on The spectrum of consistency strengths for membership in a computably enumerable set, Notre Dame Logic Seminar, April 2026
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  • Comment by Joel David Hamkins on When are two proofs of the same theorem really different proofs
    Cantor's proof does construct specific transcendental numbers explicitly. He provides an explicit construction enumerating the algebraic numbers, and then the diagonal construction produces a real number not on that list. And indeed the constructive nature of his proof was important to Cantor in his presentation.
  • Comment by Joel David Hamkins on Borel homomorphism of equivalence relation
    I edited to add the subscript 0 in the second displayed equation, since I think that is what you intended.
  • Comment by Joel David Hamkins on T=ZFC + Con(ZFC), a flawed reasoning process but why
    For a theory to prove that there is a contradiction is not the same as the theory proving a contradiction. This is the content of the second incompleteness theorem.
  • Comment by Joel David Hamkins on If $|\bigcup S|<|S|$, then there is $R\subseteq S$ so that $|\bigcup R|<|R|$ and there exists $\phi:\bigcup R\hookrightarrow R$ with $r\in\phi(r)$
    Clearly, $r\in\bigcup R$, so that $\phi(r)$ makes sense.
  • Comment by Joel David Hamkins on If $|\bigcup S|<|S|$, then there is $R\subseteq S$ so that $|\bigcup R|<|R|$ and there exists $\phi:\bigcup R\hookrightarrow R$ with $r\in\phi(r)$
    It is a very nice problem!
  • Comment by Joel David Hamkins on If $|\bigcup S|<|S|$, then there is $R\subseteq S$ so that $|\bigcup R|<|R|$ and there exists $\phi:\bigcup R\hookrightarrow R$ with $r\in\phi(r)$
    Could you clarify what theory you are working in? The usual set theories ZF, ZFC etc do not have nontrivial multisets. And you do not mention whether any version of the axiom of choice would be available.
  • Comment by Joel David Hamkins on Arbitrariness in ultrafinitism
    Oh, don't get me wrong—I totally agree with you. I have written about ultrafinitism, but my goal in the main has been just to try to understand the perspective a little better.
  • Comment by Joel David Hamkins on Arbitrariness in ultrafinitism
    @Wojowu Yes. Doron Zeilberger announced in his address at the conference that there is a largest number. But actually it seems to me that he is not working in any formal theory.

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