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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
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 Is well-orderedness by injectivity equivalent to the Axiom of Choice?
    Another subtle issue is that your formulation is not quite asserting that there is no infinite descending chain of cardinals, since by insisting that Xk+1⊆Xk you have in effect chosen an injection. So from your condition, I wouldn't necessarily know how to prove that there is no infinite descending sequence of cardinals.
  • Comment by Joel David Hamkins on Is well-orderedness by injectivity equivalent to the Axiom of Choice?
    Also, in your title you refer to well-orderedness, but already the linearity of cardinals is equivalent to the axiom of choice. Your question is closer to referring to well-foundedness, not well-orderedness, but even so, not quite, since the descending chain condition is not necessarily equivalent to well-foundedness without DC.
  • Comment by Joel David Hamkins on Is well-orderedness by injectivity equivalent to the Axiom of Choice?
    Note that your formulation is saying something like: there is no descending sequence of cardinals, but without DC this could be weaker than saying: the cardinals are well-founded, which would be: every nonempty sets of sets has one of minimal cardinality. For example, if there is an infinite Dedekind finite set, we can violate well-foundedness, […]
  • Comment by Joel David Hamkins on Is V[r0,…,rk,…] a proper extension
    If someone wants to push the idea through, please feel free to post an answer.
  • Comment by Joel David Hamkins on Is V[r0,…,rk,…] a proper extension
    Yes, in fact I was thinking about exactly that case. One needs to verify that the composition of those forcing notion is still improper. Is this clear?
  • Comment by Joel David Hamkins on Is V[r0,…,rk,…] a proper extension
    This would answer your question negatively, if we had an improper extension that was stationary preserving and generated by a real (with a Cohen real present). Perhaps we can just do some improper forcing stat-preserving forcing, followed by almost disjoint coding to get the extension generated by a real?
  • Comment by Joel David Hamkins on Is V[r0,…,rk,…] a proper extension
    If you have any Cohen real at all and W is generated by some real r, not necessary Cohen, then you can achieve the situation of your third bullet point, by splitting the Cohen real into countably many, and changing one bit in each so as to code the given real r into the sequence.
  • Answer by Joel David Hamkins for How might fundamental mathematics differ for entities with intuitive comprehension of the continuum?
    The answer to your question is the subject of descriptive set theory, which is all about trying to understand the hierarchy of logical complexity that arises in a context where the real numbers are given as basic objects. This is far beyond the arithmetic hierarchy, studying the projective hierarchy, and we analyze the complexity of […]

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