In the thread Set theories without "junk" theorems?, Blass describes the theory T in which mathematicians generally reason as follows:
Mathematicians generally reason in a theory T which (up to possible minor variations between individual mathematicians) can be described as follows. It is a many-sorted first-order theory. The sorts include numbers (natural, real, complex), sets, ordered pairs and other tuples, functions, manifolds, projective spaces, Hilbert spaces, and whatnot. There are axioms asserting the basic properties of these and the relations between them. For example, there are axioms saying that the real numbers form a complete ordered field, that any formula determines the set of those reals that satisfy it (and similarly with other sorts in place of the reals), that two tuples are equal iff they have the same length and equal components in all positions, etc.
There are no axioms that attempt to reduce one sort to another. In particular, nothing says, for example, that natural numbers or real numbers are sets of any kind. (Different mathematicians may disagree as to whether, say, the real numbers are a subset of the complex ones or whether they are a separate sort with a canonical embedding into the complex numbers. Such issues will not affect the general idea that I'm trying to explain.) So mathematicians usually do not say that the reals are Dedekind cuts (or any other kind of sets), unless they're teaching a course in foundations and therefore feel compelled (by outside forces?) to say such things.
Question: If set theorists just want to do set theory and not worry about foundations (and encodings of mathematical objects as sets), do they also work in the theory T? Or are they always regarding every object as a set?
Also, do I understand it correctly that it's hard to actually formalize the syntax of the theory T, because of the many types and connotations of natural language involved? But then, what's "first-order" about T, if T is communicated through natural language?