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Joel David Hamkins

mathematics and philosophy of the infinite

Joel David Hamkins

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Tag Archives: Irvine

How the continuum hypothesis could have been a fundamental axiom, UC Irvine Logic & Philosoph of Science Colloquium, March 2024

Posted on February 12, 2024 by Joel David Hamkins
8

This will be a talk for the Logic and Philosophy of Science Colloquium at the University of California at Irvine, 15 March 2024.

Abstract. With a simple historical thought experiment, I should like to describe how we might easily have come to view the continuum hypothesis as a fundamental axiom, one necessary for mathematics, indispensable even for calculus.

Slides-CH-could-have-been-fundamental-Hamkins-Irvine-March-2024Download

The paper is now available at How the continuum hypothesis could have been a fundamental axiom.

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Posted in Talks | Tagged continuum hypothesis, Irvine, multiverse, pluralism, thought experiment | 8 Replies

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Proof and the Art of Mathematics, MIT Press, 2020

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  • Joel David Hamkins on Lectures on Set Theory, Beijing, June 2025
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RSS Mathoverflow activity

  • Comment by Joel David Hamkins on References for incompleteness proofs using infinite trees or König's lemma
    Yes, there are diagonal arguments almost everywhere in logic.
  • Comment by Joel David Hamkins on References for incompleteness proofs using infinite trees or König's lemma
    One could object that in order to provide the tree, one must essentially have already proved the undecidability of the halting problem, and then one could prove the incompleteness theorem directly with that, using proofs to decide the halting problem, rather than doing the further work to create the computable tree with no computable branch. […]
  • Answer by Joel David Hamkins for References for incompleteness proofs using infinite trees or König's lemma
    I'm not sure about the particular proof your professor has in mind, but here is a proof using trees and paths-through-trees. First, we prove the classic result in computability theory that there is a computable infinite tree $T\subset 2^{
  • Comment by Joel David Hamkins on Where is the error in this argument against external automorphisms on models of ZFC?
    The two subset orders are isomorphic externally, by j, but there is no reason to think they are isomorphic internally, so M cannot build the map as you say. For example, even just when $j(\alpha)
  • Answer by Joel David Hamkins for Negation of cardinal characteristics
    Let me interpret your question as a request for alternative ways of thinking of the cardinal characteristics and how they relate to one another. From this perspective, the work of Corey Switzer seems relevant. In his dissertation (under my supervision), he proposed to compare the cardinal characteristics not just by comparing the least cardinal sizes […]
  • Comment by Joel David Hamkins on Does the parity obstruction hold for ∈-embeddings?
    ∈-embeddings need not take ordinals to ordinals, so it isn't just a matter of moving an ordinal up or down.
  • Comment by Joel David Hamkins on Automorphisms over models of L?
    *for any standard finite k≥1...
  • Comment by Joel David Hamkins on Automorphisms over models of L?
    And similarly, you cannot move κ to κ++ or to κ+++ or indeed to κ+k for any standard finite k, since the ordinal index would have a different value modulo k+1.

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