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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 Ordinary mathematics intrinsically requiring unbounded replacement/specification?
    Perhaps another way to think about it is: Σ2-ZFC is not that strong.
  • Comment by Joel David Hamkins on Ordinary mathematics intrinsically requiring unbounded replacement/specification?
    Σ1-ZFC is the same as bounded ZFC, when formulated with collection, since with bounded collection you can prove Σ1-collection, simply by asking for the witness of the Σ1 assertion as part of what is collected.
  • Answer by Joel David Hamkins for Ordinary mathematics intrinsically requiring unbounded replacement/specification?
    (Although this doesn't answer your question about ordinary assertions, nevertheless it seems relevant to understanding the nature of your theory.) What I claim is that your theory BZFCP is equivalent to Σ2-ZFC, that is, where we restrict the replacement axiom in ZFC to Σ2 formulas, those of the form ∀x∃y ϕ, where ϕ […]
  • Comment by Joel David Hamkins on Ordinary mathematics intrinsically requiring unbounded replacement/specification?
    Regarding UA, the relevant question would seem to be whether it counts as "ordinary" mathematics, not whether it is deemed necessary. Will you be providing a definition of what counts as ordinary? Otherwise, we might find it a moving target, like the irritating uses of "natural" one can find in similar discussions.
  • Comment by Joel David Hamkins on Ordinary mathematics intrinsically requiring unbounded replacement/specification?
    @user21820 But every Σ2 assertion is equivalent to asserting that there is some Vα satisfying a certain assertion, and so these assertions also have that absoluteness feature. See my blog post on the concept of local assertions in set theory: jdh.hamkins.org/local-properties-in-set-theory
  • Comment by Joel David Hamkins on Ordinary mathematics intrinsically requiring unbounded replacement/specification?
    One possible answer to the question is that BZFCP does not prove the existence of inaccessible cardinals, and therefore does not prove that there are any Grothendieck-Zermelo universes, which is a hypothesis in wide use outside set theory. But of course, ZFC does not prove this either. I believe that BZFCP+UA is strictly weaker than […]
  • Comment by Joel David Hamkins on Ordinary mathematics intrinsically requiring unbounded replacement/specification?
    Since the powerset operator gives you in effect another quantifier, your theory seems a bit closer to Σ2-ZFC than bounded ZFC. I wonder what the exact relation is.
  • Comment by Joel David Hamkins on Is subset-well-foundedness equivalent to the Axiom of Choice?
    I find it incorrect to say that ≤ is a partial order, and would call this instead a pre-order, since it isn't actually anti-symmetric until you take the quotient by the equivalence x≡y⟺x≤y≤x. Sets can be equinumerous, after all, without being equal.

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