set theory - first-order definability transitive closure operator

As Mike Shulman and François G. Dorais correctly point out, the official language of set theory has only the binary relation ε, and so there are no terms to speak of in that language beyond the variable symbols.

But no set theorist remains inside that primitive language, and neither is it desirable or virtuous to do so. Rather, as in any mathematical discourse, we introduce new terminology, define notions and introduce terms. What gives? I think the substance of your question is really:

• How can a set theorist (or any mathematician) sensibly and legitimately use terms that are not expressible as terms in the official language of the subject?

The answer is quite general. In any first order theory T, if one can prove that there is a unique object with a certain property, then one may expand the language by adding a term for that object, plus the defining axiom that that term has the desired property. The resulting theory T⁺ will be a conservative extension of T, meaning that the consequences of T⁺ that are expressible in the old language are exactly the same as the consequences of T. The reason is that any model M of T can be (uniquely) expanded to a model of T⁺, simply by interpreting the new term in M by its definition. This is why we may freely introduce symbols for emptyset or ω (or Q and R) and so on to set theory. Similarly, if T proves that for every x, there is a unique object y such that φ(x,y), then we may introduce a corresponding symbol f_φ(x), with the defining axiom ∀x φ(x,f_φ(x)). This new theory, in the expanded language with f_φ, is again conservative over T.

This is what is going on with the term TC(x) for the transitive closure of x. Although there is no official term for the transitive closure of x in the basic language of set theory, we may introduce such a term, once we prove that every set x does indeed have a transitive clsoure. And once having done so, the term becomes officially part of the expanded language.

To see that every set x has a transitive closure, one needs very little of ZFC, and as Dorais mentions in the comments to your question, you don't need to build the V_α hierarchy. For example, every set has a transitive closure even in models of ZF- (and much less), where the power set axiom fails and so the V_α hierarchy does not exist. Simply define a sequence x₀ = x and x_n+1 = U x_n. By Replacement, the set { x_n | n ε ω } exists, and the union of this set is precisely TC(x).

In summary, we should feel free to introduce defined terms, and there is absolutely no reason not to write TC(x) on the chalkboard, as you mentioned. In particular, we should not feel compelled to express our beautiful mathematical ideas in a primitive language with only ε, like some kind of machine code, just because it is possible in principle to do so.