Recursively saturated models of set theory

Abstract

We determine when a model M \mathfrak {M} of ZF can be expanded to a model ⟨ M , X ⟩ \langle \mathfrak {M},\mathfrak {X}\rangle of a weak extension of Gödel Bernays: GB + {\text {GB}} + the Δ 1 1 \Delta _1^1 comprehension axiom. For nonstandard M \mathfrak {M} , the ordinal of the standard part of M \mathfrak {M} must equal the inductive closure ordinal of M \mathfrak {M} , and M \mathfrak {M} must satisfy the axioms of ZF with replacement and separation for formulas involving predicates for all hyperelementary relations on M \mathfrak {M} . We also consider expansions to models of GB + Σ 1 1 {\text {GB}} + \Sigma _1^1 choice, observe that the results actually apply to more general theories of well-founded relations, and observe relationships to expansibility to models of other second order theories.