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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Recursively saturated models of set theory

Author: John Stewart Schlipf
Journal: Proc. Amer. Math. Soc. 80 (1980), 135-142
MSC: Primary 03E70; Secondary 03D70
MathSciNet review: 574523
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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.

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Keywords: Model of set theory, nonstandard model of set theory, recursively saturated model, inductive closure ordinal, hyperelementary relation, next admissible set, well-founded parts of models, $ \Delta _1^1$ comprehension, $ \Sigma _1^1$ choice
Article copyright: © Copyright 1980 American Mathematical Society

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