A Silver-like perfect set theorem with an application to Borel model theory (Q654016)

Suppose that \(L\) is a countable, recursive language. An \(L\)-prestructure is a pair \(M=(M_0,E)\) such that \(M_0\) is an \(L\)-structure and \(E\) is a congruence relation with respect to the non-logical symbols of \(L\). In this case, the quotient \(M_0/E\) is naturally an \(L\)-structure, denoted \(\tilde{M}\). We say that \(\tilde{M}\) is totally Borel over Borel if the underlying universe of \(M_0\) is \(\omega^\omega\), every symbol of \(L\) is interpreted in \(M_0\) in a Borel way, \(E\) is a Borel equivalence relation, and every definable subset of \(\tilde{M}\) is Borel. The main theorem of this paper states that an \(\omega_1\)-saturated totally Borel over Borel model of a superstable theory is saturated. The main result follows from a general perfect-set theorem concerning coanalytic notions of independence, which is of interest in its own right as is pointed out by the author via several corollaries.