Is there any known axiomatization of set theory in which the real numbers are not a set, but the natural numbers and other infinite sets do exist?
Such a set theory would have an Axiom of Infinity, but not an Axiom of Power Set. I know that Kripke-Platek set theory has no Axiom of Power Set, but it is not clear to me whether the real numbers exist as a set in this theory or not.
In the type of set theory I am envisioning, the real numbers would exist as a class, as would the class of all subsets of the natural numbers, and they could be equivalent to the class of all ordinals, for example.
One advantage of such a set theory would be that it could admit the Axiom of Choice, and the Well-Ordering Theorem for all sets, without having to admit the well-ordering of the reals or the Banach-Tarski paradox.