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Joel David Hamkins

mathematics and philosophy of the infinite

Joel David Hamkins

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Tag Archives: hyperreal numbers

How the continuum hypothesis could have been a fundamental axiom

Posted on July 3, 2024 by Joel David Hamkins
23

Joel David Hamkins, “How the continuum hypothesis could have been a fundamental axiom,” Journal for the Philosophy of Mathematics (2024), DOI:10.36253/jpm-2936, arxiv:2407.02463.

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

See also this talk I gave on the topic at the University of Oslo:

  • How the continuum hypothesis could have been a fundamental axiom, Oslo
Slides-CH-could-have-been-fundamental-Hamkins-Oslo-June-2024-1Download

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Posted in Publications | Tagged categoricity, CH, continuum hypothesis, hyperreal numbers, Leibniz, Newton, thought experiment | 23 Replies

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Proof and the Art of Mathematics, MIT Press, 2020

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  • Comment by Joel David Hamkins on Proper class sized hyperreals
    Well, of course there is no maximal one, since any such field can be made strictly larger by the same argument I describe, but I think you mean the saturated one, that is, the surreal numbers, which is unique up to isomorphism if you have global choice, and it is universal, which is a kind […]
  • Answer by Joel David Hamkins for Proper class sized hyperreals
    One can easily construct proper class sized models of any consistent theory having an infinite model. Construction using the class completeness theorem. In your case, let T be the elementary diagram of the real field $\langle\mathbb{R},+,\cdot,0,1,
  • Comment by Joel David Hamkins on Can we have nested singular sets in a Gitik model?
    Thanks. I have another question about this. Being a countable union of smaller sets and having a countable cofinality are not quite the same in the nonwellordered case. Do we know in Gitik's model that all infinite sets have countable cofinality? That is, a countable collection of strictly smaller cardinalities such that every strictly smaller […]
  • Comment by Joel David Hamkins on Can we have nested singular sets in a Gitik model?
    I usually think of Gitik's claim being that all ℵα in his model have cofinality ω, but you are claiming the stronger property that all cardinals have cofinality ω. Do we know this holds in his model? Of course, with the ℵs we get nested singularity for free.
  • Comment by Joel David Hamkins on Can we have a downshifting elementary embedding in stratified ZF?
    Is the assertion that j is elementary part of the theory? If so, could you clarify how you express that? (e.g. in ZFC context, there are choices to be made here, as with the Wholeness axiom)
  • Comment by Joel David Hamkins on Can we have a downshifting elementary embedding in stratified ZF?
    Ah, I see that now. What confused me is that you also say "work in stratified ZF", which of course usually means in the bare language of set theory. You are working in ZF(j).
  • Comment by Joel David Hamkins on Can we have a downshifting elementary embedding in stratified ZF?
    You have to specify the theory more precisely, regarding whether j is allowed into the ZF schemes. After all, we can even have full ZFC models M and isomorphisms j:M→M with $j(\alpha)
  • Comment by Joel David Hamkins on Looking for constructive example of two complementary, dense sets, both with non-zero Lebesgue measure that add to the unit interval
    You use the word "constructive," but it isn't clear whether you intend the usual informal meaning of this word in mathematics or the more particular meaning used in constructive mathematics. In the former case, there are very easy examples, such as in Jan's answer, but in the latter case, things are more subtle. After all, […]

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