A Silver-like Perfect Set Theorem with an Application to Borel Model Theory

Abstract

A number of results have been obtained concerning Borel structures starting with Silver and Friedman followed by Harrington, Shelah, Marker, and Louveau. Friedman also initiated the model theory of Borel (in fact totally Borel) structures. By this we mean the study of the class of Borel models of a given first-order theory. The subject was further investigated by Steinhorn. The present work is meant to go further in this direction. It is based on the assumption that the study of the class of, say, countable models of a theory reduces to analyzing a single $\omega_1$-saturated model. The question then arises as to when such a model can be totally Borel. We present here a partial answer to this problem when the theory under investigation is superstable.