Set theoretical analogues of the Barwise-Schlipf theorem

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Abstract

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 αOrdM.

(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 αOrdM.

(b) There is X such that (M,X)GB+Δ11-CA+Σ11-AC.

MSC

primary
03C62
03E30
secondary
03C70
03H99

Keywords

Zermelo-Fraenkel set theory
Gödel-Bernays class theory
Recursive saturation
Nonstandard models
Infinitary language
Forcing

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