Ralph Loader writes: It's perfectly feasible to imagine that in say, descriptive set theory, properties of certain structures could be shown by showing (1) they have a certain cardinality, (2) sets of such cardinality have epsilon trees with a certain property and (3) using epsilon induction over such epsilon trees to prove something about our original structures. [I don't know of any such instances; I'm not a descriptive set theorist.] There are all sorts of near-by examples but they're not counterexamples to my assertion. The fact that any structure of a certain type is isomorphic to one with further special properties is a standard first-step. The most usual further property is the existence of a well-ordering. But there's no such example, nor will there ever be, in which one must use some property of the elements of every possible structure of the given type in order to prove a property of the structure. That's a tautology if the "property of the structure" in question is an isomorphism-invariant. Alternatively, what about Godel's constructible universe. This seems to be a mathematical construction, and the actual element relation seems to be not only of interest, but essential to the construction. I must agree that when making constructions in mathematical subjects in which the elements are the essence then one will use elements. About my example of the philosopher who asked the number theorist (I actually witnessed this) whether he was proving theorems about Russell's or Van Neumann's numbers led Ralph to reply: You state a question in mathematical English, and then criticize ZF for being able to express this question, while category theory cannot. One wonders what other questions stated in mathematical English---some of them perhaps perfectly sensible---can be stated in the language of ZF, but not in the language of category theory. In fact there are topos-theoretic analogues for Russell and Van Neumann, but not even a philosopher would be tempted to ask the analogue question. Besides the R and VN numbers one could ask the analogue question about the Lawvere and the Church numbers (in category theory, not in set theory). But one wouldn't. Anyway, I do have a counterexample to my own assertion. I think JH Conway gave us some examples of mathematical constructions in which the elements are the essence, to wit, his games and his numbers. Conway games can be described as the result of replacing the single epsilon of ZF with a pair of epsilons (for left and right "moves"). If one restricts attention to "impartial" games (any move legal for one player would have been legal for the other) then the two epsilons can be conflated and the subject conflates to ZF.