Has any open/difficult problem in ordinary mathematics been solved only/mostly by appeal to set theory?

Question

We know that many (if not all) mathematical notions can be reduced to the talk of sets and set-membership. But it nevertheless sounds like a grueling task (if at all possible) to actually get advanced results in many branches of ordinary mathematics if we only work with sets and set-membership relation in our language, or otherwise only rely on set theory. To put it differently: it seems that in order to get results in many branches of mathematics one might not need to be very familiar with set theory at all, let alone being able to translate everything to the language of sets or to heavily rely on set theory.

I'm wondering if there are cases where an open/a difficult problem in other branches of mathematics (e.g., number theory or real analysis) has been solved mostly/only because of the insight that set theory has offered, directly or indirectly (say, through branches that heavily appeal to set theory, such as model theory). Even a historical incident will be helpful: a problem of the sort that was first solved thanks to set theory, but later on more accessible solutions have been found that don't deal much with sets.

Thank you very much!

Answer 1

I suspect you'll get a pretty wide range of answers here. There are lots of examples of questions arising in not-set-theory that have turned out to be independent of ZFC. Here's another example that I'm quite fond of, which has a different flavour, in that set-theoretic methods gave an outright answer to a question isn't obviously about set theory.

Let $X$ be a Polish space, and let $B_1(X)$ be the space of (real-valued) Baire class $1$ functions on $X$; that is, functions which can be obtained as the pointwise limit of a sequence of continuous functions. Give $B_1(X)$ the topology of pointwise convergence. Todorcevic proved that every compact subspace of $B_1(X)$ contains a dense metrizable subspace, answering a question that had been raised in functional analysis. His proof uses set theory in a very deep way. As far as I know, no one has found a proof that doesn't involve heavy set-theoretic machinery.

Since your question mentioned model theory, let me also mention Hrushovski's proof of the relative Mordell-Lang conjecture in positive characteristic. His proof used model theory to solve a question arising from number theory. The way that model theory is used in the proof isn't especially set-theoretic, but much of the machinery he used originated in a part of model theory (Shelah's classification theory) that does have strong interactions with set theory.

Answer 2

Shelah's black box is used widely in solving algebra problems. One example that I like is the following work of Dugas and Göbel

All infinite groups are Galois groups over any field. Trans. Amer. Math. Soc. 304 (1987), no. 1, 355–384.

In this paper, Shelah's black box is used to prove the infinite analogue of the still unsolved Hilbert-Noether inverse Galois problem.

Also a nice and short reference is The uses of set theory by Roitman. The following is taken from Mathscinet:

The author's purpose is to show how modern set theory is relevant to other parts of mathematics, particularly areas not ordinarily regarded as close to set theory (in contrast to, e.g., general topology). After a brief section on set-theoretic background, most of the paper consists of specific examples of connections between set theory and other areas. Two of the examples are mentioned only very briefly, because thorough expositions of them exist elsewhere. These are the independence of Kaplansky's conjecture on automatic continuity of certain Banach space homomorphisms and Whitehead's conjecture about freeness of abelian groups. The other six examples are presented in somewhat more detail, including the basic ideas behind the proofs, in approximately one-half to one (large) page per example.

• The first example concerns the work of G. Weiss, S. Shelah, and the reviewer, relating properties of ideals of compact operators on Hilbert space to the combinatorial principle of near coherence of filters and establishing the consistency and independence of this principle.
• The second is a characterization by J. Steprāns of free abelian groups as those admitting discrete norms.
• The third is Shelah's theorem that the fundamental group of a nice space is either finitely generated or of the cardinality of the continuum.
• The fourth is an independence result arising in strong homology theory, where a result proved by S. Mardešić and A.V. Prasolov under the continuum hypothesis was shown by A. Dow, P. Simon, and J. Vaughan to be unprovable in ZFC.
• The fifth is an example, due to Shelah and Steprāns, of a nonseparable Banach space where every linear operator is a scalar multiplication plus an operator with separable range.
• The last concerns R. Laver's work, arising from large cardinal theory, on the free left-distributive algebra on one generator. The paper includes references to either the original sources or surveys for each of the examples presented. (P. Dehornoy has recently shown that the irreflexivity of the ordering in the author's last example can be proved without large cardinal hypotheses; yet another connection with mainstream mathematics is exhibited by the title of Dehornoy's as yet unpublished paper: "Braid groups and left distributive structures''.)

Answer 3

Here are two examples:

1. Say an abelian group has a norm if, well, it has a function which behaves like a norm. We say this norm is discrete if the range of this norm in the real numbers is discrete. Clearly every free abelian group has a discrete norm. What about the other direction? It turns out that indeed, if. $G$ has a discrete norm, it is a free abelian group. This was shown by Juris Steprāns in

   Steprāns, Juris, A characterization of free Abelian groups, Proc. Am. Math. Soc. 93, 347-349 (1985). ZBL0566.20037.

2. Let $G$ be an uncountable group, does it have an uncountable subgroup other than itself? If $G$ is abelian, yes. But it turns out that there is a non-abelian example of an uncountable group such that every proper subgroup is countable. This was shown by Saharon Shelah in

   Shelah, Saharon, On a problem of Kurosh, Jonsson groups, and applications, Word problems II, Stud. Logic Found. Math. Vol. 95, 373-394 (1980). ZBL0438.20025.

Answer 4

Dehornoy order in braid theory was first discovered using large cardinal axioms. For a while it was an open problem if large cardinal axioms were actually required, although since then elementary approaches have been discovered.

Answer 5

Even a historical incident will be helpful: a problem of the sort that was first solved thanks to set theory, but later on more accessible solutions have been found that don't deal much with sets.

I think Bernstein and Robinson's solution to the invariant subspace problem for polynomially compact linear operators on a Hilbert space may qualify for this. Bernstein and Robinson are making use of non-standard analysis in their proof, which is the insight provided by logic as you say. Later on, Halmos published a "classical" proof which, he states in abstract, is a modification of Bernstein and Robinson argument.

Answer 6

Quoting https://en.wikipedia.org/wiki/Ax%E2%80%93Kochen_theorem,

"The Ax–Kochen theorem, named for James Ax and Simon B. Kochen, states that for each positive integer $d$ there is a finite set $Y_d$ of prime numbers, such that if $p$ is any prime not in $Y_d$ then every homogeneous polynomial of degree $d$ over the $p$-adic numbers in at least $d^2 + 1$ variables has a nontrivial zero.

"The proof of the theorem makes extensive use of methods from mathematical logic, such as model theory."