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They announced that they had a proof the next year, in 2012. Woodin immediately identified a mistake, and they withdrew their paper in shame. They revisited the proof often in the years that followed, but they invariably found that they lacked one key idea — a “missing link,” Asperó said, in the logical chain leading from Martin’s maximum++ to (*).
Their plan of attack for deriving the latter axiom from the former was to develop a forcing procedure similar to L-forcing with which to generate a type of object called a witness. This witness verifies all statements of the form of (*). So long as the forcing procedure obeys the requisite consistency condition, Martin’s maximum++ will establish that the witness, since it can be forced to exist, exists. And thus (*) follows.
“We knew how to build such forcings,” Asperó said, but they couldn’t see how to guarantee that their forcing procedure would meet the key requirement of Martin’s maximum. Asperó’s “flash experience” in the car in 2018 finally showed the way: They could break up the forcing into a recursive sequence of forcings, each satisfying necessary conditions. “I remember feeling very confident that this ingredient would in fact make the proof work,” he said, though it took further flashes of insight from both Asperó and Schindler to work it all out.
The convergence of Martin’s maximum++ and (*) creates a solid foundation for a tower of infinities in which the cardinality of the continuum is ℵ2. “The question is, is it true?” asks Peter Koellner, a set theorist at Harvard.
According to Koellner, knowing that the strongest forcing axiom implies (*) can count as evidence either for or against it. “Really that depends on what your take on (*) is,” he said.
The convergence result will focus scrutiny on (*)’s plausibility, since (*) allows mathematicians to make those powerful “for all X, there exists Y” statements that have consequences for the properties of the real numbers.
Despite (*)’s extreme usefulness in permitting those statements, seemingly without contradiction, Koellner is among those who doubt the axiom. One of its implications — a mirroring of the structure of a certain large class of sets with a much smaller set — strikes him as strange.
Notably, the person who might have been most enthusiastic about (*)’s correctness has also turned against it. “I’m considered a traitor,” Woodin said in one of our Zoom conversations this summer.
Twenty-five years ago, when he posed (*), Woodin thought the continuum hypothesis was false, and thus that (*) was a source of light. But about a decade ago, he changed his mind. He now thinks that the continuum has cardinality ℵ1 and that (*) and forcing are “doomed.”
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.
Gödel’s incompleteness theorems.
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.”
Clarification July 23, 2021:
This article was changed to eliminate a possible misimpression about the number of order types of natural numbers.
They announced that they had a proof the next year, in 2012. Woodin immediately identified a mistake, and they withdrew their paper in shame. They revisited the proof often in the years that followed, but they invariably found that they lacked one key idea — a “missing link,” Asperó said, in the logical chain leading from Martin’s maximum++ to (*).
Their plan of attack for deriving the latter axiom from the former was to develop a forcing procedure similar to L-forcing with which to generate a type of object called a witness. This witness verifies all statements of the form of (*). So long as the forcing procedure obeys the requisite consistency condition, Martin’s maximum++ will establish that the witness, since it can be forced to exist, exists. And thus (*) follows.
“We knew how to build such forcings,” Asperó said, but they couldn’t see how to guarantee that their forcing procedure would meet the key requirement of Martin’s maximum. Asperó’s “flash experience” in the car in 2018 finally showed the way: They could break up the forcing into a recursive sequence of forcings, each satisfying necessary conditions. “I remember feeling very confident that this ingredient would in fact make the proof work,” he said, though it took further flashes of insight from both Asperó and Schindler to work it all out.
The convergence of Martin’s maximum++ and (*) creates a solid foundation for a tower of infinities in which the cardinality of the continuum is ℵ2. “The question is, is it true?” asks Peter Koellner, a set theorist at Harvard.
According to Koellner, knowing that the strongest forcing axiom implies (*) can count as evidence either for or against it. “Really that depends on what your take on (*) is,” he said.
The convergence result will focus scrutiny on (*)’s plausibility, since (*) allows mathematicians to make those powerful “for all X, there exists Y” statements that have consequences for the properties of the real numbers.
Despite (*)’s extreme usefulness in permitting those statements, seemingly without contradiction, Koellner is among those who doubt the axiom. One of its implications — a mirroring of the structure of a certain large class of sets with a much smaller set — strikes him as strange.
Notably, the person who might have been most enthusiastic about (*)’s correctness has also turned against it. “I’m considered a traitor,” Woodin said in one of our Zoom conversations this summer.
Twenty-five years ago, when he posed (*), Woodin thought the continuum hypothesis was false, and thus that (*) was a source of light. But about a decade ago, he changed his mind. He now thinks that the continuum has cardinality ℵ1 and that (*) and forcing are “doomed.”
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.
Gödel’s incompleteness theorems.
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.”
Clarification July 23, 2021:
This article was changed to eliminate a possible misimpression about the number of order types of natural numbers.