Do set theorists work in T? 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?
 A: Caveat number 1: strictly speaking, no one actually works in the theory $T$, just as no one actually works in the theory $\mathsf{ZFC}$. Mathematicians work by means of carefully used natural language and not within a formal system. Formal systems are formulated as approximations that try to model what mathematicians actually do while at work.
Now to address the question, with the above caveat in mind, are we always regarding every object as a set? Not necessarily always, just sometimes. The point is that $\mathsf{ZFC}$ and $T$ are bi-interpretable, so you can switch between both viewpoints at will without that changing the stuff that you can prove (and even better: both $T$ and $\mathsf{ZFC}$ are just approximations to what we actually do, so we can just do math as usual, and not worry about these nuances, and whatever it is that we're doing can in theory be translated to the formal system of your choice).
A: You address your question to set theorists, and so let me answer as
a set theorist that, yes, when I think purely as a set theorist,
then indeed the idea that every object is a set just goes without
saying — it is a basic elementary part of the ZFC conceptual
framework. One needn't ever even remark on this in an argument made
to other set theorists.
For example, the cumulative $V_\alpha$ hierarchy provides an
extremely rich structural background for the set-theoretic
universe, which is especially informative and helpful in the
analysis of various set-theoretic axioms, especially the large
cardinal axioms that reach very high into this hierarchy. The
picture of all objects existing as sets in the cumulative hierarchy
is basically pervasive in set-theoretic arguments, and is
definitely a fundamental part of the set-theoretic understanding of
the mathematical universe. One may freely carry out an argument by
$\in$-induction, for example, concluding that every object $x$ has
a certain property, although really what one has proved is that
every well-founded set has the property. In this sense, I would say
that when set theorists are operating as set theorists amongst set
theorists, they are not usually operating in the theory T described
by Andreas, but rather in something much closer to ZFC, in a
language expanded by concepts that have been defined in set theory. (As David mentioned, esentially no mathematician, including set theorists, works in a purely formal system.)
Meanwhile, however, this doesn't mean that set-theorists don't make
use of type-theoretic concepts. For example, set theorists have
diverse concepts of what counts as a real number, and one can
commonly find various real-number concepts used in set theory,
including: an element of Cantor space $r\in 2^\omega$; a subset
of the natural numbers $r\subseteq\omega$; an element of Baire
space $r\in \omega^\omega$; and so on. Often the algebraic
field-theoretic structure of the reals is less important or
relevant to set-theoretic concerns, and set theorists typically
care about real numbers as: a package containing countably much
information. In many set-theoretic contexts, it doesn't matter
which particular real-number concept one is using, and in this
sense talk about the reals reduces essentially to the use of a
real-number type.
Finally, however, when set theorists communicate with other
mathematicians, then of course they are naturally able to
communicate in something much closer to the theory $T$ that you
mentioned.
