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

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

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.

**[1]**A. Blass,*The model of set theory generated by countable many generic reals*, J. Symbolic Logic**46**(1981), 732-752. MR**641487 (83e:03050)****[2]**H. Friedman,*Large models with countable height*, Trans. Amer. Math. Soc.**20**(1975), 227-239. MR**0416903 (54:4966)****[3]**W. Guzicki,*Generic families and models of set theory ZFC*(to appear).**[4]**R. Solovay,*A model of set theory in which every set of reals is Lebesgue measurable*, Ann. of Math.**92**(1970), 1-56. MR**0265151 (42:64)****[5]**Z. Szczepaniak and A. Zarach,*Consistency of the theory ZFC*+*"every set has a Hartogs number"*+*"continuum is a proper class"*+*GCH*, Instytut Matematyki Politechniki Wroclawskiej, Wroclaw, 1978.

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

Article copyright:
© Copyright 1989
American Mathematical Society