An interesting post on divine infinity at "RazorsKiss". By coincidence, I've recently been thinking of this issue. As I've noted before, Malebranche makes a big deal about infinity; it plays an immensely important role in a number of key arguments. (I recently got a search engine hit from someone looking for Malebranche's argument for the existence of God; Malebranche's argument is based on infinity.) Necessity, immutability, eternity, and universality all show up, of course; but they are treated as subordinate and derivative properties.
But the arguments generally assume that only God is infinite. Now, I think it is quite right that only God is absolutely infinite -- infinite in His very being, we might say. But I'm not at all convinced that creatures are incapable of being infinite; my inclinations are toward the view expressed by Aquinas, that creatures can be relatively infinite. Indeed, on Aquinas's view it seems to follow that every creature actually is relatively infinite. Material things are materially infinite, and forms not completely contracted by matter are formally infinite. Material infinity is not particularly impressive; it is a potential infinitude rather than an actual infinitude (I use 'infinitude' rather than 'infinity' because 'infinity' suggests a real collection of some sort, which is not quite what Aquinas is getting at). The interesting creaturely infinitude is formal infinitude, because if Aquinas is right, Malebranche is wrong in thinking that we must be ontologists, i.e., we must hold that ideas are actually in God; or, at least, Malebranche's major argument to this effect is wrong. A scholastic can answer Malebranche's challenge by denying the assumption that most clearly allows one to get around the problem he notes. (I suggested this possibility before, but only speculatively; I hadn't thought to look at Aquinas in this connection.)
On the other hand, if either Aquinas or Malebranche are correct about this matter, no purely naturalistic account of the mind is possible; the naturalist must argue against Aquinas's view that the soul is formally infinite, but because of that must find another answer to Malebranche's challenge to provide a non-question-begging account of our ability to recognize that things are infinite. (For my more analytically-minded readers, Malebranche's arguments are somewhat similar to Thomas Nagel's arguments about the infinite in The Last Word, with somewhat different emphases; the two sets of arguments actually complement and strengthen each other quite well.) Malebranche, I think, completely demolishes the typical, and rather unimpressive, empiricist responses to which people usually flee. Empiricists can't get a genuine infinite. Berkeley recognized this in his notebooks, which is why he is forced to conclude there that the Pythagorean theorem is a useful calculating device that, strictly speaking, is false (because it can only be true at infinite precision).
