Recursively saturated models of set theory
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- by John Stewart Schlipf PDF
- Proc. Amer. Math. Soc. 80 (1980), 135-142 Request permission
Abstract:
We determine when a model $\mathfrak {M}$ of ZF can be expanded to a model $\langle \mathfrak {M},\mathfrak {X}\rangle$ of a weak extension of Gödel Bernays: ${\text {GB}} +$ the $\Delta _1^1$ comprehension axiom. For nonstandard $\mathfrak {M}$, the ordinal of the standard part of $\mathfrak {M}$ must equal the inductive closure ordinal of $\mathfrak {M}$, and $\mathfrak {M}$ must satisfy the axioms of ZF with replacement and separation for formulas involving predicates for all hyperelementary relations on $\mathfrak {M}$. We also consider expansions to models of ${\text {GB}} + \Sigma _1^1$ 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.References
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Additional Information
- © Copyright 1980 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 80 (1980), 135-142
- MSC: Primary 03E70; Secondary 03D70
- DOI: https://doi.org/10.1090/S0002-9939-1980-0574523-6
- MathSciNet review: 574523