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Set theoretical analogues of the Barwise-Schlipf theorem
We characterize nonstandard models of ZF (of arbitrary cardinality) that can be expanded to Goedel-Bernays class theory plus $Δ^1_1$-Comprehension. We also characterize countable nonstandard models of ZFC that can be expanded to Goedel-Bernays class theory plus $Σ^1_1$-Choice.
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