How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer - II

Subhotosh Khan

Super Moderator
Staff member
Jun 18, 2007
(Continued from How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer - I.)

To address this problem, mathematicians posed various “forcing axioms” — rules that established the actual existence of specific objects rendered possible by Cohen’s method. “If you can imagine an object to exist, then it does; this is the guiding intuitive principle that leads to forcing axioms,” Magidor explained. In 1988, Magidor, Matthew Foreman and Saharon Shelah took this ethos to its logical

conclusion by posing Martin’s maximum, which says that anything you can conceive of using any forcing procedure will be a true mathematical entity, so long as the procedure satisfies a certain consistency condition.

For all the expansiveness of Martin’s maximum, in order to simultaneously permit all those products of forcing (while satisfying that constancy condition), the size of the continuum jumps only to a conservative ℵ2— one cardinal number more than the minimum possible value.

Besides settling the continuum problem, Martin’s maximum has proved to be a powerful tool for exploring the properties of infinite sets. Proponents say it fosters many sweeping statements and general theorems. By contrast, assuming that the continuum has cardinality ℵ1 tends to yield more exceptional cases and roadblocks to proofs — “a paradise of counterexamples,” in Magidor’s words.

Martin’s maximum became massively popular as an extension of ZFC. But then in the 1990s, Woodin proposed another compelling axiom that also kills the continuum hypothesis and pins the continuum at ℵ2 but by a totally different route. Woodin named the axiom (*), pronounced “star,” because it was “like a bright source — a source of structure, a source of light,” he told me.

(*) concerns a model universe of sets that satisfies the nine ZF axioms plus the axiom of determinacy, rather than the axiom of choice. Determinacy and choice logically contradict each other, which is why (*) and Martin’s maximum seemed irreconcilable. But Woodin devised a forcing procedure by which to extend his model mathematical universe into a larger one that is consistent with ZFC, and it’s in this universe that the (*) axiom holds true.

What makes (*) so illuminating is that it lets mathematicians make statements of the form “For all X, there exists Y, such that Z” when referring to properties of sets within the domain. Such statements are powerful modes of mathematical reasoning. One such statement is: “For all sets of ℵ1 reals, there exist reals not in those sets.” This is the negation of the continuum hypothesis. Thus, according to (*), Cantor’s conjecture is false. The fact that (*) lets mathematicians conclude this and assert many other properties of sets of reals makes it an “attractive hypothesis,” Schindler said.

With two highly productive axioms floating around, proponents of forcing faced a disturbing surplus. “Both the forcing axiom [Martin’s maximum] and the (*) axiom are beautiful and feel right and natural,” Schindler said, so “which one do you choose?”

If the axioms contradicted each other, then adopting one would mean sacrificing the other’s nice consequences, and the judgment call might feel arbitrary. “You would have had to come up with some reasons why one of them is true and the other one is false — or maybe both should be false,” Schindler said.

Instead, his new work with Asperó shows that Martin’s maximum++ (a technical variation of Martin’s maximum) implies (*). “If you unify these theories, as we did,” Schindler said, “I would say that you can take it as a case in favor of: Maybe people got something right.”

Missing Link Asperó and Schindler were young researchers together at an institute in Vienna 20 years ago. Their proof germinated several years later, when Schindler read a manuscript, handwritten as usual, by the set theorist Ronald Jensen. In it, Jensen invented a technique called L-forcing. Schindler was impressed by it and asked a student of his to try to develop it further. Five years later, in 2011, he described L-forcing to Asperó, who was visiting him in Münster. Asperó immediately suggested that they might be able to use the technique to derive (*) from Martin’s maximum++.

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.

Other Stars 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.”