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**A Silver-like perfect set theorem with an application to Borel model theory.**
*(English)*
Zbl 1247.03052

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.

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.

Reviewer: Isaac Goldbring (Los Angeles)