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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Generic families and models of set theory with the axiom of choice


Authors: Krzysztof Ciesielski and Wojciech Guzicki
Journal: Proc. Amer. Math. Soc. 106 (1989), 199-206
MSC: Primary 03C62; Secondary 03E25
MathSciNet review: 994389
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Abstract: Let $ M$ be a countable transitive model of ZFC and $ A$ be a countable $ M$-generic family of Cohen reals. We prove that there is no smallest transitive model $ N$ of ZFC that either $ M \cup A \subseteq N$ or $ M \cup \{ A\} \subseteq N$. It is also proved that there is no smallest transitive model $ N$ of ZFC$ ^{-}$ (ZFC theory without the power set axiom) such that $ M \cup \{ A\} \subseteq N$. It is also proved that certain classes of extensions of $ M$ obtained by Cohen generic reals have no minimal model.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1989-0994389-8
PII: S 0002-9939(1989)0994389-8
Article copyright: © Copyright 1989 American Mathematical Society