I describe several ways in which forcing arguments can be used to yield clean and conceptual proofs of nonreducibility, ergodicity and other results in the theory of analytic equivalence relations. In particular, I present simple Borel equivalence relations $E, F$ such that a natural proof of nonreducibility of $E$ to $F$ uses the independence of the Singular Cardinal Hypothesis at $\aleph_\omega$.
"Analytic equivalence relations and the forcing method." Bull. Symbolic Logic 19 (4) 473 - 490, December 2013. https://doi.org/10.2178/bsl.1904020