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:
Complement: If the sheep can be determinately isolated from the non-sheep, then the non-sheep can be determinately isolated from the sheep.
Separation: If the sheep can be determinately isolated from the non-sheep, then the black sheep can be determinately isolated from the non-black-sheep.
["non-black-sheep" here refers to anything that is not a black sheep, not anything that is a non-black sheep!]
The immediate answer that I usually get is that the intuition behind complement, but not the intuition behind separation, depends on the collection of all objects - the universal set - being determinate (so if the sheep are a determinate collection, than everything 'left over' is as well). But this strikes me as a little question-begging. After all, it is only our acceptance of separation (and of ZFC more generally) that throws the existence of the universal set into question. There are all sorts of alternative set theories (such as New Foundations or Boolos' DualV) that accept complement (and most of the standard ZFC axioms in the case of DualV) and hence accept the existence of a universal set.
So, what arguments can be given for the claim that it is separation and not complement that we should retain - arguments that don't depend on things like the iterative conception of set that already presuppose rejection of complement?