TITLE: Examples of set theory problems which are solved using methods outside of logic
QUESTION [10 upvotes]: 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?

REPLY [8 votes]: Inspired by Joel David Hamkin's comment---Simon Thomas has provided applications of various super-ridigity theorems (from the ergodic theory of group actions) to the theory of the Borel complexity of countable equivalence relations, for example he shows that the universal countable equivalence relation is not essentially free 
Thomas, Simon, Popa superrigidity and countable Borel equivalence relations, Ann. Pure Appl. Logic 158, No. 3, 175-189 (2009). ZBL1162.03029.