(Ockinf) For any X's, there is some y such that y is not one of the X's.which, for a suitable definition of infinity, gives us an infinitely large universe, but no infinitely large Ockham set to correspond to it. Every Ockham set is finite, yet the universe is infinite. As 'Belette' has spotted, this may be no surprise, for it is well known that the axiom of infinity cannot be derived from the rest of the axioms of Zermelo Fraenkel set theory, and we can even construct a model of the axioms where the axiom of infinity is replaced by its negation (called ZF-INF).
But it is not so simple as that, and I will complete my argument by returning to the point I made in my first post on this subject, about the difference between Ockham sets and mathematical ones. The key difference is that, given the existence of the elements corresponding to an Ockham set, the set itself has to exist, for the set is identical with its members. This is Malezieu's principle: if twenty men exist, it is because the first, the second, the third man etc., exist. If a exists and b exists, then the Ockham set a and b exists also. By contrast, in set theory we have to posit this using an axiom such as the axiom of pairs.
Thus, given a universe with an infinite number of elements but no oset corresponding to them, can we wave a magic wand and bring that corresponding oset into existence, as we can easily do in ordinary set theory? I suggest we can't. For an oset simply is its members. If we bring an oset into existence, we must bring at least one of its elements into existence. But we can't do that here. By hypothesis, every member of the infinitely large universe exists already. What is lacking is the oset of all of them. If we now create the oset, what exactly are we creating? Nothing. There are no things to create, for they are already there. Nor can we suppose that there is any other possible universe corresponding to this possible one, containing identical elements, but also containing the oset itself. So, if a universe consisting of these elements is possible, it is necessarily the case that there is no oset containing all its elements. And in general, there can be no universe containing an infinite number of elements, together with the oset of all of them. If every possible world is such that p, then necessarily p.
This is a surprising result. The alternative is to reject (Ockinf) above, and thus reject the possibility of an infinite universe without a corresponding oset. But that is equally surprising, for it would amount to rejecting the independence of the axiom of infinity. We could merely define 'finite oset' as we did before, assert that to every finite oset there is an (equally finite) oset containing one extra element, and then deduce the existence of an infinite oset, based on the logical impossibility of (Ockinf) above.