Are you saying the the set of all finite sets fails to satisfy ZF axioms?
Let S denote the "set of all sets", then there is a one-to-one mapping fromOriginally Posted by ThePerfectHacker
S into S1 the "set of all sets with one element", namely the map f which takes
s in S to {s} in S1. Now consider S1'=f(S). Then the map f is one-to-one and
onto form S to S1'.
Now f is also a one-to-one map from the power set P(S) of S onto the Power
Set of P(S1') of S1'. But if S1 is a set so is S1' as it is a subset of S1, so the
cardinality of P(S1') is strictly greater than the cardinality of S1'. But this
would imply that the cardinality of P(S) is strictly greater than that of S, but
its not so it aint.
RonL