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 M is a subset of N and either A is subset of N or A is an element of 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 is a subset of N and A is an element of N. It is also proved that certain classes of extensions of M obtained by Cohen generic reals have no minimal model.

Full text on line in pdf format. Requires Adobe Acrobat Reader.

Last modified January 5, 2002.