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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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  • [J] Barwise [1975], Admissible sets and structures, Springer-Verlag, Berlin. MR 0424560 (54:12519)
  • [J] Barwise and J. Schlipf [1976], On recursively saturated models of Peano arithmetic, Model Theory and Algebra: A Memorial Tribute to Abraham Robinson (D. H. Saracino and B. Weispfenning, Eds.), Lecture Notes in Math., vol. 498, Springer-Verlag, Berlin, pp. 42-54. MR 0409172 (53:12934)
  • [K] Bielinski [1977], Extendability of structures as an infinitary property, Set Theory and Hierarchy Theory. V (A. Lachlan, M. Srebrny, and A. Zarach, Eds.), Lecture Notes in Math., vol. 619, Springer-Verlag, Berlin, pp. 75-94. MR 0469694 (57:9475)
  • [A] Ehrenfeucht and G. Kreisel [1966], Strong models of arithmetic, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 14, pp. 107-110. MR 0219415 (36:2497)
  • [U] Felgner [1971], Comparison of the axioms of local and universal choice, Fund. Math. 71, pp. 43-62. MR 0289285 (44:6476)
  • [H] Gaifman [1975], Global and local choice functions, Israel J. Math. 22, pp. 257-265. MR 0389590 (52:10421)
  • [L] Harrington, L. Kirousis and J. Schlipf [1978], A generalized Kleene-Moschovakis theorem, Proc. Amer. Math. Soc. 68, pp. 209-213. MR 0476457 (57:16020)
  • [H] J. Keisler [1973], Forcing and the omitting types theorem, Studies in Model Theory (M. D. Morley, Ed.), Math. Assoc. of Amer., Buffalo, N. Y., pp. 96-133. MR 0337571 (49:2340)
  • [G] Kreisel [1965], Axioms for explicitly definable properties, Appendix A to Informal rigour and completeness proofs, Problems in the Philosophy of Mathematics (Imre Lakatos, Ed.), North-Holland, Amsterdam, pp. 162-165.
  • [Y] N. Moschovakis [1971], Predicative classes, Axiomatic Set Theory, Part 1, Proc. Sympos. Pure Math., vol. 131, Amer. Math. Soc., Providence, R. I., pp. 247-264. MR 0281599 (43:7314)
  • 1. -[1974], Elementary induction on abstract structures, North-Holland, Amsterdam.
  • [J] P. Ressayre [1977], Models with compactness properties relative to an admissible language, Ann. Math. Logic 11, pp. 31-55. MR 0465849 (57:5735)
  • [J] Schlipf [1977], A guide to the identification of admissible sets above structures, Ann. Math. Logic 12, pp. 151-192. MR 0485330 (58:5177)
  • 2. -[1978], Toward model theory through recursive saturation, J. Symbolic Logic 43, pp. 183-206. MR 0491130 (58:10399)

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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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