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 -CA (the scheme of -Comprehension). In what follows, , is the set of formulae of the infinitary logic that appear in the well-founded part of , and -AC is the scheme of -Choice.
Theorem A. The following are equivalent for a nonstandard model of ZFC of any cardinality:
(a) for an unbounded collection of .
(b) -CA, where is the family of -definable subsets of .
(c) There is such that -CA.
Theorem B. The following are equivalent for a countable nonstandard model of ZFC:
(a) for an unbounded collection of .
(b) There is such that .