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.

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**Last modified January 5, 2002.**