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 be a countable transitive model of *ZFC* and be a countable -generic family of Cohen reals. We prove that there is no smallest transitive model of *ZFC* that either or . It is also proved that there is no smallest transitive model of *ZFC* (*ZFC* theory without the power set axiom) such that . It is also proved that certain classes of extensions of obtained by Cohen generic reals have no minimal model.

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DOI:
https://doi.org/10.1090/S0002-9939-1989-0994389-8

Article copyright:
© Copyright 1989
American Mathematical Society