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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 How "natural" is Paris-Harrington?
    To my way of thinking, we should rather ground our notions of "natural" in mathematical features than in sociological ones. For example, perhaps we should say that properties are natural when they exhibit certain kinds of absoluteness or uniformity or preservation and so on. The difficulty is to find the right notions that align in […]
  • Comment by Joel David Hamkins on Seeking a nontrivial first-order theory T with Mod(T) being linear under embeddability
    Wojciech, very nice. Your final example provides a theory T for which Mod(T) is linearly preordered by embeddability, but not model complete. But yet, it is not a counterexample to S4.3 for assertions with parameters, which as you know was the motivation for the question, since I believe that S4.3 is still valid here. However, […]
  • Comment by Joel David Hamkins on How "natural" is Paris-Harrington?
    In my view, "natural" is a mathematically empty concept, a proxy for other properties, not yet articulated or formulated. The mathematical questions here would be: does the PH result have this specific mathematical regularity feature or that one? One must do the work of articulating a specific notion of naturality. Without that, it seems meaningless. […]
  • Comment by Joel David Hamkins on Seeking a nontrivial first-order theory T with Mod(T) being linear under embeddability
    @AliEnayat That is an interesting idea, which I had tried to use, but it doesn't actually work as I need. I proved that the countable models of set theory are linearly preordered by embeddability, as relational structures. But this isn't true for uncountable models, and so Mod(ZFC) is not linearly preordered via embeddings.
  • Comment by Joel David Hamkins on Seeking a nontrivial first-order theory T with Mod(T) being linear under embeddability
    The application is in modal model theory, and with QE we get modality trivialization, which prevents the counterexample from working. Meanwhile, we have now found a way to make the counterexample work with a comparatively trivial toy theory. But I would still be interested in examples for this question.
  • Comment by Joel David Hamkins on Seeking a nontrivial first-order theory T with Mod(T) being linear under embeddability
    Examples of theories where Mod(T) is linear would include the theory of equality and the theory of algebraically closed fields of characteristic p. But these both also have QE. Slightly less trivial is the theory of acyclic graphs where every point has degree 2. But this has QE in the language with distance-k, which is […]
  • Comment by Joel David Hamkins on Seeking a nontrivial first-order theory T with Mod(T) being linear under embeddability
    I am also extremely interested in the possibility that that the situation is impossible. Please post a proof, which will be warmly accepted.
  • Comment by Joel David Hamkins on Seeking a nontrivial first-order theory T with Mod(T) being linear under embeddability
    @NoahSchweber Yes, the theory needs to be trivial in some respects, but the question is whether that is possible in other respects.

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