Joel David Hamkins, “How the continuum hypothesis could have been a fundamental axiom,” Journal for the Philosophy of Mathematics (2024), 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.

The JPM will launch in September 2024. Meanwhile, the preprint pdf is available at arxiv.org/pdf/2407.02463.

See also this talk I gave on the topic at the University of Oslo:

I agree with Solovay that ~CH could also have been seen as fundamental via an intuition for the existence of a countably additive real valued measure.

But there’s an important difference. None of the alternative foundational schemes you treat in this paper buy us anything new in the arithmetical realm (or more generally anything covered by Shoenfield absoluteness), unless you want to make it second-order and get a little bit further to Grothendieck universes. But taking RVM as fundamental has far more in the way of concrete consequences, up to consistency of measurables for arithmetical sentences and settling most of the wall-known descriptive set theory questions left open by ZFC.

It’s also interesting to take this in the other direction and ask what alternative historical developments could have led us to adopt axioms inconsistent with ZFC. This is well-covered ground in the case of alternatives to AC, but those don’t get us anything concrete (Shoenfield absoluteness again). But what developments might have made ZF implausible?

Thesis: no conceivably plausible alternative historical development of mathematics would contradict ZF about any *arithmetical* statements.

This seems reasonable, although it is an implicit commitment to Con(ZF), which some have doubted. For example, Silver conjectured the negation, and someone with that view would support, say, PA + not Con(ZF). We know that this is consistent relative to ZF, so one can’t really object to the basic coherence of it. Certainly it is plausible. And if one had a strong form of it, denying consistency at the level of $\Sigma_{1000}$-replacement, for example, then this would be an arithmetic statement directly contradicting ZF.

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Can infinity have a measure or adegree of approximation? or we shall leave it abstract as mathematicians!!

Perhaps the consideration of demonstrability of the continuum hypothesis could have led mathematical sciences along a different path of development: https://doi.org/10.1007/s10699-022-09875-9

I agree with you. CH should have been taken as the most fundamental axiom in mathematics. For historical reasons (we arrived at CH through countable numbers), we have the present view. Countable numbers are arbitrary units. If we cut ‘one’ candy into hundred pieces, we can say that we have hundred candies, only the unit ‘one’ differs. The unit ‘one’ can be either extremely small or extremely large, always finite, but can never be infinitesimal or infinite. The unit is just an arbitrary finite piece of the continuum.

But in physics (my interest is in theoretical physics), ‘quantum hypothesis’ is the most fundamental axiom (in my opinion). Matter is quantized, and so universe made up of such units is finite. That is the fundamental difference between mathematics and physics.

Does ‘countably saturated’ mean that the set of hyperreals between any two real numbers has cardinality aleph-0?

No, it means that every countably defined gap is filled. That is, if $x_0\leq x_1\leq x_2\leq\cdots y_2\leq y_1\leq y_0$ with $x_i