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Joel David Hamkins

mathematics and philosophy of the infinite

Joel David Hamkins

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Tag Archives: Leibniz

How the continuum hypothesis could have been a fundamental axiom

Posted on July 3, 2024 by Joel David Hamkins
30

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 | 30 Replies

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  • Comment by Joel David Hamkins on Confusion regarding the requirements for a recursive ordinal notation
    @AndreasBlass Right, I think we are in agreement. In my experience, these days when people say "computable ordinal" they just mean that the order relation is computable, but when people talk about a "denotation system" they probably want more. Meanwhile, I take the constructions I described to show (1) one may always have the extra […]
  • Comment by Joel David Hamkins on Confusion regarding the requirements for a recursive ordinal notation
    By multiplying by $\omega^2$ or more, one can also make the "next limit" operation computable, and much more.
  • Comment by Joel David Hamkins on Confusion regarding the requirements for a recursive ordinal notation
    @AndreasBlass In terms of which ordinals are represented, those requirements give the same ordinals. If I have a computable well-ordered relation, after all, I can make a computable relation with your properties simply by muliplying it by omega, adding a new $\omega$ chain above each point. In the new order, limits and successors are also […]
  • Answer by Joel David Hamkins for Countinuous "refinement" of a function $f : [\omega]^\omega \to [\omega]^\omega$
    I guess the metric space you have in mind for $[\omega]^\omega$ takes the difference of two infinite sets as a weighted sum of the symmetric difference, where the weights converge. For example, $d(A,B)=|A\bigtriangleup B|$, where $|A|=\sum_{n\in A}1/2^{n+1}$ for $A,B\subseteq\omega$. This space has a countable dense subset, consisting of the cofinite sets, and so there are […]
  • Comment by Joel David Hamkins on Is there an “opposite” hypothesis to the (Generalized) Continuum Hypothesis?
    @JorgenHarmse Yes, in set theory we have an amazing control over the values of the beths, expressed by Easton's theorem. And yes, MPccc is an axiom scheme.
  • Comment by Joel David Hamkins on Apophatic mathematics
    In classical logic, to assert p is the same as asserting not not p, so isn't every foundational theory apophatic?
  • Comment by Joel David Hamkins on Is the set theoretic multiverse view akin to the idea in proof theory that different logics are worth studying in their own right?
    I'm not sure what kind of answer is wanted. Isn't it obvious that there is at least a superficial resemblance here between logical pluralism and set-theoretic pluralism? I have met many people who didn't realize at first that these were distinct topics. But in truth, these topics have largely distinct communities of researchers, even if […]
  • Comment by Joel David Hamkins on Predicates of infinite arity
    @TaylorRendon Usually the terminology is that a finitary relation means a relation that is $n$-ary for some finite $n$. It would be sensible to have a relation of variable finitary arity, allowing all finite arities, for example, but this isn't how the terminology is usually used.

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