I shall be giving a keynote lecture for the CFORS Grad Conference at the University of Oslo, 19-20 June 2024.
Abstract. I shall describe a simple historical thought experiment showing how our attitude toward the continuum hypothesis could easily have been very different than it is. If our mathematical history had been just a little different, I claim, if certain mathematical discoveries had been made in a slightly different order, then we would naturally view the continuum hypothesis as a fundamental axiom of set theory, one furthermore necessary for mathematics and indeed, indispensable for calculus.
The paper is now available at How the continuum hypothesis could have been a fundamental axiom.
And if certain mathematical and physical discoveries had been made in another different order, we’d be accepting the existence of a real-valued measure on all subsets of the continuum as a fundamental axiom, and that contradicts the continuum hypothesis.
That could very well be true, and to the extent it is successful this serves in support of my main point, which is that there is historical contingency in what counts as a fundamental axiom. In particular, let me emphasize that I am not proposing my thought experiment as any kind of defense of CH, but rather only to point out how we might have had very different attitudes about what counts as a fundamental axiom.
Meanwhile, I think I’ve heard your thought experiment on Twitter, and I just don’t find it as compelling or believable as my example, so I’m not convinced by it. Nevertheless, as a historical point in support of your view, I’ve heard Solovay say that he strongly believes that the continuum is real-valued measurable.
Wonderful. Are the Cohen Reals a hyperreal field? Seemingly yes. So doesn’t this demonstrate non-categoriity of hyperrreals without CH ? Jerome Tauber
The Cohen reals do not form a field. Nevertheless, it is true that without CH, there are multiple non-isomorphic countably saturated fields of size continuum, even though this is a categorical characterization when CH holds.
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