Recursively saturated models of set theory

Schlipf, John Stewart

doi:10.1090/S0002-9939-1980-0574523-6

Recursively saturated models of set theory
HTML articles powered by AMS MathViewer

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

Jon Barwise, Admissible sets and structures, Perspectives in Mathematical Logic, Springer-Verlag, Berlin-New York, 1975. An approach to definability theory. MR 0424560
Jon Barwise and John Schlipf, On recursively saturated models of arithmetic, Model theory and algebra (A memorial tribute to Abraham Robinson), Lecture Notes in Math., Vol. 498, Springer, Berlin, 1975, pp. 42–55. MR 0409172
Konrad Bielinski, Extendability of structures as infinitary property, Set theory and hierarchy theory, V (Proc. Third Conf., Bierutowice, 1976) Lecture Notes in Math., Vol. 619, Springer, Berlin, 1977, pp. 75–93. MR 0469694
A. Ehrenfeucht and G. Kreisel, Strong models of arithmetic, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 14 (1966), 107–110 (English, with Russian summary). MR 219415
Ulrich Felgner, Comparison of the axioms of local and universal choice, Fund. Math. 71 (1971), no. 1, 43–62. (errata insert). MR 289285, DOI 10.4064/fm-71-1-43-62
Haim Gaifman, Global and local choice functions, Israel J. Math. 22 (1975), no. 3-4, 257–265. MR 389590, DOI 10.1007/BF02761593
Leo Harrington, Lefteris Kirousis, and John Schlipf, A generalized Kleene-Moschovakis theorem, Proc. Amer. Math. Soc. 68 (1978), no. 2, 209–213. MR 476457, DOI 10.1090/S0002-9939-1978-0476457-5
H. Jerome Keisler, Forcing and the omitting types theorem, Studies in model theory, MAA Studies in Math., Vol. 8, Math. Assoc. Amer., Buffalo, N.Y., 1973, pp. 96–133. MR 0337571

1965

Axioms for explicitly definable properties

Informal rigour and completeness proofs

Yiannis N. Moschovakis, Predicative classes, Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) Amer. Math. Soc., Providence, R.I., 1971, pp. 247–264. MR 0281599

1974

Elementary induction on abstract structures

J. P. Ressayre, Models with compactness properties relative to an admissible language, Ann. Math. Logic 11 (1977), no. 1, 31–55. MR 465849, DOI 10.1016/0003-4843(77)90009-2
John S. Schlipf, A guide to the identification of admissible sets above structures, Ann. Math. Logic 12 (1977), no. 2, 151–192. MR 485330, DOI 10.1016/0003-4843(77)90013-4
John Stewart Schlipf, Toward model theory through recursive saturation, J. Symbolic Logic 43 (1978), no. 2, 183–206. MR 491130, DOI 10.2307/2272817