TITLE: Is there a Second-Order Axiomatization of ZF(C) which is categorical?
QUESTION [9 upvotes]: A theory $T$ is called categorical if it only has one model upto isomorphism.  (Note: this has nothing to do with category theory.)  The Lowenheim-Skolem theorem states that no first-order theory with an infinite model is categorical.  However, a second-order theory CAN be categorical, although such a theory can't be recursively axiomatizable (because there's no recursive axiomatization of second-order logic that is complete with respect to standard semantic).
For instance, the Peano axioms with the full second-order induction axiom comprise a second-order version of first-order Peano arithmetic, and they constitute a categorical axiomatization of the natural numbers.  Similarly, the axioms for ordered fields, together with the Dedekind completeness axiom (every bounded set has a least upper bound), comprise a second-order version of the first-order theory of real closed fields, and they constitute a categorical axiomatization of the real numbers.
My question is, can we do the same thing for set theory?  That is to say, is there a second-order version of ZF, or ZFC, which is categorical?
Any help would be greatly appreciated.
Thank You in Advance.

REPLY [2 votes]: Van Mcgee, in his "How We Learn Mathematical Language" gives an argument for the categoricity of ZFCU (ZFC with Urelements) plus the axiom that the Urelements form a set (the "Urelement Set Axiom"). This is obviously not quite what you were asking about, but perhaps is close enough to be of interest.
The "Categoricity Theorem" is given thusly:

Categoricity Theorem. Any two models of second-order ZFCU + the Urelement Set Axiom with the same universe of discourse have isomorphic pure sets. In particular, any two models of second-order ZFCU + the Urelement Set Axiom in which the first- order variables range over everything have isomorphic pure sets.

He does, however, note a sense in which this system is not categorical. In his footnote 33 he says:

Sometimes a specification of a mathematical system is regarded as categorical only if it characterizes the system, uniquely up to isomorphism, by its internal structure. Our axioms are not categorical is this strict sense, since we characterize the pure sets by reference to things that aren't pure sets, namely, the Urelemente. Usage here is not uniform. Hilbert's axioms for geometry are often referred to as "categorical," even though they refer to things outside the given system of points, lines, and planes.

So, I suppose whether you'd accept this as an example of a categorical system depends on how precisely you use "categorical".