For any set x and predicate F(...), there is a set y such that,
for all z: z is a member of y iff (z is a member of x and F(z))
and of complement: For any set x, there is a set y such that,
for all z: z is a member of y iff z is not a member of x. are
incompatible with the existence of any sets. Proof: Assume a set
exists. Then by separation, the empty set exists. So by
complement the universal set exists. So by separation, the
Russell set of all non-self-membered sets exists. Russell's
paradox follows. So, we need to give up either separation or
complement. Needless to say, standard mathematical practice has
opted for the latter option. The question is: Should we have? In
other words, are there philosophically respectable reasons for
thinking that it is complement, and not separation, that should
be jettisoned? Each of these principles corresponds to a natural
intuition to have about the 'definiteness' or 'determinateness'
of the results of certain set-forming operations. To put things
in a bit silly terms: