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?
The constructivists/intuitionists like Brouwer and Weyl and Heyting win the philosophical war in the early 20th century and excluded middle itself is viewed with suspicion by mainstream mathematicians. People instead work in IZF + dependent/countable choice or some similar set theory.
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
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Hi Joel. VeryHi Joel. Very interesting paper. I’d like to raise a possibly new objection.
Simply: What about the Levi-Civita field? https://en.wikipedia.org/wiki/Levi-Civita_field
This is defined as the field of finitely-supported formal power series $\sum_{q \in \mathbb{Q}} a_q \varepsilon^q$ with real coefficients $a_q$. It is *not* countably saturated, and of course $\omega$-cofinal. However, it has several virtues, including being real-closed and (astonishingly, to me) useful in computational analysis (see the wikipedia article and links inside).
Additionally, it is characterized by a second-order categorical theory. This just piggybacks on the categoricity of $\mathbb R$ and $\mathbb N$.
Now, I don’t know if it has all the advantages of a hyperreal field. But being real-closed, it at least is elementarily equivalent to $\mathbb R$. One can imagine that this relatively concrete field, if it were conceived by early analysts, might have sufficed for the infinitesimal applications they were after.
Levi-Civita is only one among many fields that extend the reals but are not given by an ultrapower. But it seems like a particularly elegant one.
Thanks for the comment, and I’m glad you like the paper. That field is concrete enough that one can imagine it coming very early, and perhaps giving substance to the infinitesimal ideas. For my argument, however, I don’t think I am obligated to show that the solution I describe is the only possible thing that could have happened (since it didn’t in fact happen that way in any case), but rather my task was only to argue that it *could* have happened in the way I describe in my thought experiment. And I think I have done that much, so have explain how it could have been that CH is viewed as a fundamental axiom, necessary for mathematics.
I see, thanks. I agree that you’ve successfully argued that the scenario you described *could have* played out that way. But my impression was that the claim was a bit stronger, that if the early analysts had been blessed with rigor about number systems, the hyperreals would have been the most natural thing to use, leading to CH as a way to secure categoricity. I think examples like Levi-Civita, given their simplicity (in hindsight, admittedly) mitigate the sense of inevitability of the alternate history.
I think it could be plausibly argued that the notion of countable saturation may have been too abstract for early analysts, and maybe the concreteness of Levi-Civita would have won more favor, if there had been a contest between them.