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.

The paper is now available at How the continuum hypothesis could have been a fundamental axiom.

Will this talk (of great interest to me) be available on line?

charlie sitler

charlies@beekmanschool.org

I’m not sure, but I shall find out and report back here about that.

The continuum hypothesis, and its intimations regarding infinity, I find FASCINATING!

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.

Dropping continuity seems like a rather large change.

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).

I just happened on this and found it very interesting! Your description of how things happened in your imaginary world is pretty convincing, but I’m not sure I agree with your conclusions about how forcing would be perceived in the imaginary world. From my perspective, what set theorists call “forcing” is really just a small fragment of a general family of constructions that in general don’t preserve AC or even LEM, namely sheaves/forcing over arbitrary sites. Set theorists in our world generally restrict themselves to forcing over posets with the double-negation topology *because* that sort of forcing does preserve LEM and AC, and ZFC is their “fundamental theory”. So it seems to me that in your imaginary world, with ZFC+CH as the “fundamental theory”, set theorists would probably simply redefine “forcing” to mean what today we would call “forcing that preserves CH”, leaving it to the maverick topos theorists to explore worlds in which CH, AC, and LEM fail.

I agree completely with this, and I think this is something that I hint at in the article (now available at https://jdh.hamkins.org/how-ch-could-have-been-fundamental). I think the imaginary world attitude toward forcing with arbitrary complete Boolean algebras would be something like current attitudes in set theory toward symmetric extensions, which don’t always preserve the axiom of choice.