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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 Does every dense down-set in P(ω)/(fin) contain a partition?
    It is not a cBa, since no countably infinite antichain has a supremum, since by Hausdorff's method you can squeeze another set below any candidate. Every (ω,ω) gap is filled.
  • Comment by Joel David Hamkins on Does every dense down-set in P(ω)/(fin) contain a partition?
    This won't imply DC, since one can have AC for all sets at this level, and a violation of DC only very high up in the set-theoretic universe.
  • Comment by Joel David Hamkins on Extracting partitions from dense open subsets of complete Boolean algebras without choice
    Conceivably it is easier to find antichains and partitions in complete Boolean algebras than in partial orders. For example, every antichain in a cBa extends to a partition by taking the complement of the supremum. Do we know the question for partial orders is equivalent to the question for complete Boolean algebras?
  • Comment by Joel David Hamkins on Extracting partitions from dense open subsets of complete Boolean algebras without choice
    Calliope, you don't just want to say D≠{0}, but rather require a,b≠0 in your definition of density.
  • Comment by Joel David Hamkins on Ordinary mathematics intrinsically requiring unbounded replacement/specification?
    @user21820 I believe Caleb's point is that precisely because every Π10 statement is equivalent to a consistency statement, it is not actually so easy to have a robust conception of "ordinary" mathematics that excludes the kinds of issues that logicians investigate. Indeed, I find the category at bottom as meaningless and irritating as "natural". But […]
  • Comment by Joel David Hamkins on Ordinary mathematics intrinsically requiring unbounded replacement/specification?
    There is a whole literature on this.
  • Comment by Joel David Hamkins on Ordinary mathematics intrinsically requiring unbounded replacement/specification?
    I don't view that kind of justification as different in kind from the essentially similar story that people with the universe view tell about the cumulative hierarchy going all the way up in a totally deterministic manner, getting ZFC and much more, including large cardinals, on philosophical reflection grounds. In both cases, the justifications proceed […]
  • Comment by Joel David Hamkins on Ordinary mathematics intrinsically requiring unbounded replacement/specification?
    Well, no theory can be justified to be sound in a non-circular way, except by appealing to a stronger theory in the metatheory, which one might naturally take to show that the object theory should have been stronger to begin with.

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