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.
Best Answer
There is an important set-theoretic issue here: to be "categorical", a theory must have only one "model". If some reasonable candidate for second-order ZFC was categorical, its unique "model" would have to be the class of all sets. But then it would not have a set model, so it would actually be inconsistent in second-order semantics.
Put another way: there is a cardinal $\kappa$ such that if a countable theory in second-order logic has any model, then it has a model of size less than or equal to $\kappa$ (this is related to the Löwenheim number of second-order logic). But it would not make sense to call a set theory "second order ZFC" if its unique model had size less than $\kappa$, since we know there are sets larger than $\kappa$. And no matter what countable second-order theory we consider, we will never manage to exceed $\kappa$. (Surely any reasonable candidate for second-order ZFC would have at most a countable number of axioms.) So most of the set-theoretic universe would be omitted by any candidate for "categorical second-order ZFC".
Nevertheless, there are theories that are often called "second order ZFC". One such theory is just ZFC, but with the axiom scheme of replacement replaced by a single second-order axiom that quantifies over every class function $f$, and says that the image of any set under any class function is again a set. These theories are not categorical in second order logic, but at least they are consistent, and their models are much more nicely behaved than arbitrary models of first-order ZFC.