Set theoretical analogues of the Barwise-Schlipf theorem

Abstract

We characterize nonstandard models of ZF (of arbitrary cardinality) that can be expanded to Goedel-Bernays class theory plus $\Delta^1_1$-Comprehension. We also characterize countable nonstandard models of ZFC that can be expanded to Goedel-Bernays class theory plus $\Sigma^1_1$-Choice.