How Many Numbers Exist?-III

Subhotosh Khan

Super Moderator
Staff member
Jun 18, 2007
(Continued from How Many Numbers Exist?-II
Woodin called Asperó and Schindler’s proof “a fantastic result” that “deserves to be in the Annals” — the Annals of Mathematics is widely considered to be the top math journal — and he acknowledged that this kind of convergence result “is usually taken as evidence of some kind of truth.” But he doesn’t buy it. There’s the issue mentioned by Koellner, and another even bigger problem that he identified in a flash experience of his own in 2019, shortly after reading the preprint of Asperó and Schindler’s paper. “It’s an unexpected twist in the story,” Woodin said.

When he posed (*), Woodin also posed stronger variants called (*)+ and (*)++, which apply to the full power set (the set of all subsets) of the reals. It’s known that, in various models of the mathematical universe if not in general, (*)+ contradicts Martin’s maximum. In a new proof, which he began to share with mathematicians in May, Woodin showed that (*)+ and (*)++ are equivalent, which means (*)++ contradicts Martin’s maximum in various models also.

(*)+ and (*)++ far outshine (*), for one reason: They permit mathematicians to make statements of the form “There exists a set of reals …” and thus to describe and analyze properties of any and all sets of reals. (*) does not provide such an “existential theory” of sets of reals. And because Martin’s maximum seems to contradict (*)+ and (*)++, it seems that existential statements about sets of reals might not be possible in the Martin’s maximum framework. For Woodin, this is a deal breaker: “What this is saying is, it’s doomed.”

The other main players are all still digesting Woodin’s proof. But a few stressed that his arguments are conjectural. Even Woodin acknowledges that a surprising discovery could change the picture (and his opinion), as has happened before.

Many in the community await the results of Woodin’s attempt to prove the “ultimate L” conjecture: that is, the existence of an all-encompassing generalization of Gödel’s model universe of sets. If ultimate L exists — Woodin has good reason to think it does, and he is 400 pages into a proof attempt now — he’ll consider it obvious that the “dream axiom” to add to ZFC must be the ultimate L axiom, or the statement that ultimate L is the universe of sets. And in ultimate L, Cantor is right: The continuum has cardinality ℵ1. If the proof works out, the ultimate L axiom will be, if not an obvious choice of extension for ZFC, at least a formidable rival for Martin’s maximum.

Ever since Gödel and Cohen established the independence of the continuum hypothesis from ZFC, infinite math has been a choose-your-own-adventure story in which set theorists can force the number of reals up to any level — ℵ35, or ℵ1000, say — and explore the consequences. But with Asperó and Schindler’s result pointing compellingly to ℵ2, and Woodin building the case for ℵ1, a clear dichotomy has established itself, and an outright winner seems newly possible. Most set theorists would like nothing more than to exit the mathematical multiverse and coalesce behind a single picture of Cantor’s paradise, one that’s beautiful enough to call true.

Kennedy, for one, thinks we may soon return to that “preLapsarian world.” “Hilbert, when he gave his speech, said human dignity depends upon us being able to decide things in mathematics in a yes-or-no fashion,” she said. “This was a matter of redeeming humanity, of whether mathematics is what we always thought it was: to establish the truth. Not just this truth, that truth. Not just possibilities. No. The continuum is this size, period.”