In this article on Skolem's Paradox, it gives the Power Set Axiom as an example where models may badly misinterpret the axioms (compared to the "intention" of the axiom). While originally I thought that the axiom guaranteed the presence of all subsets of a set, in fact it just seems to guarantee that there's a set containing whichever subsets happen to be present in a model.
My understanding is that because of the Löwenheim–Skolem theorem, we'll never be able to come up with an axiom which guarantees that a model will have e.g. all "actual" subsets of $\mathbb{N}$. It seems unfortunate that we can never get models to behave as we intend.
Are there other axioms which still do a better job than the Power Set Axiom at requiring models to contain subsets of each of their sets? What about when we consider other logical systems, or maybe allow proofs of infinite length?