Set theoretical analogues of the Barwise-Schlipf theorem
Ali Enayat · Annals of Pure and Applied Logic · 2022
We prove the following characterizations of nonstandard models of ZFC (Zermelo-Fraenkel set theory with the axiom of choice) that have an expansion to a model of GB (Gödel-Bernays class theory) plus Δ11-CA (the scheme of Δ11-Comprehension). In what follows, M(α):=(V(α),∈)M, LM is the set of formulae of the infinitary logic L∞,ω that appear in the well-founded part of M, and Σ11-AC is the scheme of Σ11-Choice. Theorem A. The following are equivalent for a nonstandard model M of ZFC of any cardinality: (a) M(α)≺LMM for an unbounded collection of α∈OrdM. (b) (M,X)⊨GB+Δ11-CA, where X is the family of LM-definable subsets of M. (c) There is X such that (M,X)⊨GB+Δ11-CA. Theorem B. The following are equivalent for a countable nonstandard model of ZFC: (a) M(α)≺LMM for an unbounded collection of α∈OrdM. (b) There is X such that (M,X)⊨GB+Δ11-CA+Σ11-AC.