set theory - Ordinals that are not sets

Yes. This is both studied by set theorists and interesting. I personally find some of the related questions below extremely interesting, connected with some very deep questions about the nature of mathematical existence.

While the term "ordinal" is usually used only to refer to objects that are sets, one can nevertheless consider class relations that are well-ordered. For example, it is easy to define a relation on ORD that has order type ORD+1; one can make order type ORD+ORD by defining a relation that puts all even ordinals before all odd ordinals, and so on. One can get on a long way with this method, past ORD^ORD, ORD^ORD^ORD, and so on, simply by defining more complicated well-ordered relations on ORD. The flavor here is something like the manner that one considers computable well-ordered relations on the natural numbers, and defines the class of computable ordinals. The analogy is indeed very strong, for the supremum of the computable ordinals is known as ω₁^CK, pronounced "omega 1 Church Kleene", and this is precisely the smallest admissible ordinal, which means that the corresponding level of the constructible universe satisfies the Kripke-Platek axioms of set theory, a weak fragment of ZFC.

Similarly, for the situation with the definable class well-ordered relations on ORD, if one looks at the corresponding (constructible) universe of sets that arises from these super-ordinals, it also can be admissible. Another way to describe the situation is that models of ZFC can extend to models of KP, by adding precisely the sets on top coded by a class well-founded relation on ORD.

The question looming in the background here, is the extent to which every model of set theory is an initial segment of a much taller model of set theory. Can we extend every model of ZFC to an initial segment of a taller model of ZFC? Of ZFC-? Of KP?

Part of the answer is that every model of Kelly-Morse set theory arises similar to this way. Namely, every model of KM is the V_alpha of a model of ZFC-. (Edit: one may need additional technical hypotheses here.) Perhaps this is part of the reason why Kelly-Morse set theory is rarely studied in set theory, since it reduces to ZFC in this way. To have a model of KM is simply to have a model of ZFC that is the V_alpha of a model of ZFC-.

Another part of the answer is that every countable computably-saturated model of ZFC is precisely the V_alpha of another model of ZFC, to which it is externally isomorphic. In particular, every such model of ZFC is (externally) isomorphic to a rank initial segment of itself.

Another part of the answer is that every model of ZFC elementarily embeds into another model, which is the V_alpha of a much taller model. This can be proved by an easy compactness argument. Given any model M of ZFC, write down the theory consisting of ZFC+the elementary diagram of M, relativized to V_delta, in the language with an additional constant delta. This theory is finitely realized by M itself, on account of the Reflection Theorem, and so it has a model N. The original model M embeds into V_delta^N because of the elementary diagram, so N is as desired.

Finally, note that if M has the property that it is point-wise definable (every object is definable in M without parameters), and there are definitely such models of ZFC if there are any at all, then M cannot be the V_alpha of any model N of ZFC, since N would recognize that M is pointwise definable, and thus N would have only countably many reals, a contradiction.

Let me point out that Harry's observation (now apparently removed) about Grothendieck universes amounts to a special case of the question I pose, where one considers only transitive models of set theory, under the assumption that there is a proper class of inaccessible cardinals (which is equivalent to the Universe axiom). In this case, obviously every transitive model of set theory extends to one of the V_kappa, for kappa inaccessible above, and this is essentially what he observed. But actually, one doesn't need this strength to make the conclusion, since the large cardinal consistency strength of the assertion that every transtive set is an element of a larger transitive model of set theory is strictly weaker than even one inaccessible cardinal, let alone a proper class of them.