How the continuum hypothesis could have been a fundamental axiom, UC Irvine Logic & Philosoph of Science Colloquium, March 2024

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.

6 thoughts on “How the continuum hypothesis could have been a fundamental axiom, UC Irvine Logic & Philosoph of Science Colloquium, March 2024

  1. I have an alternate story, how the negation of CH could have become a fundamental axiom!

    If general relativity had been developed before quantum mechanics, when “continuous matter” was still a possibility, people would have found the hypothesis of a real-valued measure on all subsets of the continuum plausible, and after working with it for decades, would have a strong intuition for it being consistent, so that the Banach-Tarski theorem would have led them to drop the requirement of rotation invariance rather than that of continuous space (general relativity is fine with anisotropy). Godel’s results on L would merely be considered to indicate that ZFC was too weak, and the power of RVM would have been even more appreciated because it implies Con(ZFC) by taking L(kappa) for weakly inaccessible kappa. All large cardinals below measurables, after Solovay’s work, would be regarded as unproblematic.

      • That makes my story more plausible. Banach-Tarski showed that some of our naive intuitions had to go—because it was after both general relativity and quantum theory, either continuity of matter or isotropy of space were candidates and mathematicians chose to give up on continuity. But if they had worked with RVM more, perhaps if Ulam’s results had been found earlier, by the time the crisis came they would have had enough intuition that it was harmless and consistent that they would have kept it because it was so powerful (proving Con(ZF) and not-CH and much more besides).

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