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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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  • 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, […]
  • Comment by Joel David Hamkins on Simple true Π10 statements independent of weak arithmetics
    I don't think that terminology is universal, as I have heard people refer to IΣn as a weak arithmetic, and I would find it odd to describe PA as a "strong" arithmetic. But I think this may be a US/Europe variation. Also, you are more connected with the weak part of the topic than I […]
  • Comment by Joel David Hamkins on Simple true Π10 statements independent of weak arithmetics
    @EmilJeřábek You don't regard PRA as a weak arithmetic? I have always taken the boundary to be drawn at PA—anything weaker than PA is a weak arithmetic. (And not only me.) How does the terminology break down in your usage?
  • Comment by Joel David Hamkins on Simple true Π10 statements independent of weak arithmetics
    @C7X That's a better question. For Andres, my point is that separation over extremely weak theories is cheap and easy—why should we take this as revealing anything important or meaningful? Many incontrovertible assertions are independent over weak theories.
  • Comment by Joel David Hamkins on Simple true Π10 statements independent of weak arithmetics
    You ask for independence over "some much weaker" theory, but of course independence over a theory implies independence over all weaker theories, and any nontrivial statement is independent over any sufficiently weak theory. So of course we can find very low-number states with independence of the kind you seek over a sufficiently weak theory. But […]

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