(Ockinf) For any x's, there is some y such that y is not one of the x's.Against: if we can speak of 'any x' in an infinite domain, then surely we can speak of 'all the x's'. But if the axiom above is true, it follows that we cannot speak of all x's.
For: there is no logical impossibility here. Indeed, if we take x's as being an ordinary mathematical set, the statement corresponds to a version of standard set theory (being part of ZF-inf). Why should a different in the interpretation of the terms lead to a difference in truth value?
So I shall leave this one for now (but any comments or ideas gratefully accepted). It's connected with a wider question of whether there could be a nominalist version of set theory, and whether it would differ in any way from standard set theory.