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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.

Keshav Srinivasan
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2 Answers2

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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.

Carl Mummert
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  • It's important to point out that "class" doesn't mean what it usually means (a collection which is first-order definable over the model), but rather "any subcollection of the universe". – Asaf Karagila Sep 21 '13 at 18:55
  • As a matter of opinion, I would say that "class" normally does mean an arbitrary subcollection of the universe, and the use of "class" to mean "definable class" is idiosyncratic to some treatments of ZFC. – Carl Mummert Sep 21 '13 at 18:59
  • Hmm.. I must have been treating $\sf ZFC$ like that for quite some time, because I can't recall the last time there was any distinction... :-) – Asaf Karagila Sep 21 '13 at 19:30
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    The second-order replacement axiom is strictly stronger than the second-order separation axiom. Consider cardinals $\kappa$ such that $V_\kappa$ is a model of (first-order) ZFC. If there is an inaccessible cardinal $\lambda$, then the first such $\kappa$ is strictly smaller than the first such $\lambda$. Then $V_\kappa$ satisfies second-order separation but not second-order replacement (because $\kappa$ actually has cofinality $\omega$). – Andreas Blass Sep 21 '13 at 19:34
  • @Andreas: thanks - I managed to assume replacement in my mental proof that the two were equivalent. – Carl Mummert Sep 21 '13 at 19:37
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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".

Dennis
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