# Examples of set theory problems which are solved using methods outside of logic

The question is essentially the one in the title.

Question. What are some examples of (major) problems in set theory which are solved using techniques outside of mathematical logic?

• Getting tenure? :) – Asaf Karagila Aug 15 '18 at 8:08
• Can you give an example problem for which this could conceivably happen? I have a feeling that if someone asks, for example, a consistency question about Banach spaces, and then it turns out to have a ZFC answer via methods internal to functional analysis, then we would just say it wasn't a set-theoretical problem after all. – Monroe Eskew Aug 15 '18 at 8:35
• More of a speculation than an answer: It's conjectured that every Suslin ccc forcing adds a Cohen real or a random real. Shelah showed that if such a forcing adds an unbounded real then it adds a Cohen real. Farah and Zapletal showed that if $\mathbb P$ is Suslin ccc and $\omega^{\omega}$-bounding then $RO(\mathbb P)$ is a Maharam algebra. Therefore, the above problem reduces to the following: Does every Maharam algebra adds a random real? Talagrand showed that there is a Maharam algebra which is not a measure algebra, solving an old problem by Von Neumann... – Haim Aug 15 '18 at 19:55
• I don't think logic or set theory have such sharp boundaries that it is possible to provide a definitive answer. Huge parts of set theory, such as Borel equivalence relation theory or set-theoretic topology, are deeply connected with other related areas, and it could sometimes be difficult to describe a method as existing in only set theory or the companion area. – Joel David Hamkins Aug 17 '18 at 22:38
• If you had asked for logic problems outside of logic, rather than set theory problems outside of set theory, then maybe Tarski's problems about the first order of free groups theory fit the bill, having been solved by methods of geometric group theory. – Lee Mosher Aug 17 '18 at 22:43