Of course I don't believe that set theory is the One True Foundation and everything else is a lie, the fact that one can give a foundation with just one type of object, just one binary relation and relatively few simple axioms (or axiom schemas) is quite relaxing and I would say a bit unappreciated.

And also unlike other fellow students I never encountered any problem with more seemingly complicated constructions like tensor products or free groups since one can easily see how they are coded in set theory if one is familiar with it as a foundation.

The advantage is, then, that we can use a simple first order logic, where all objects in the logic are of the same type. This makes certain things easier and more pleasant. That the proposition `1 ∈ 2` can be written (i.e. that it is not a syntax error, though it's value is unknowable) should not bother us, just as that the English proposition "the sky is Thursday" is not a grammatical error and yet is nonsensical, doesn't bother us. It is no more or less bothersome than being able to write the proposition `1/x = 13`, with its result remaining equally "undefined" (i.e. unknowable and uninteresting) if x is 0. If `1/x = 13` isn't a syntax error, there's no reason `1 ∈ 2` must be a syntax error, either.

That a proposition is nonsensical (for all assignments of variables or for some specific ones, as in x = 0 in 1/x) need not be encoded in the grammar of the logic at all, and defining nonsense as "unknowable and uninteresting" is both convenient and elegant. I think that some logicians overlook this because they're attracted to intuitionist theories, where the notion of provability is more reified, whereas in classical theories every proposition is either true or false. They're bothered perhaps less by the ability to write 1 ∈ 2 and more by the idea that 1 ∈ 2 has a truth value. But while the notion of provability itself is not reified in classical logics, unprovable propositions are natural and common. 1 ∈ 2 has a meaning only in a very abstract sense; the theory can make that statement valid yet practically nonsensical by not offering axioms that can prove or disprove it. Things can be "undefined" in a precise way: the axioms do not allow you to come to any definition.

Indeed, this is exactly how the formal set theory in TLA+ is defined: https://pron.github.io/posts/tlaplus_part2

In mathematics labels are _not_ important, definitions are.

One simple example that everybody can relate to: do natural numbers include 0 or not? Who cares? Some definitions include it, some do not. There's even a convention of using N for N with 0, and N+ for excluding it, but even the convention is just a convention, not a definition. You could call them "funky numbers, the set of whole positive numbers including 0", and you're fine. Funky, natural, those are just names, labels, as long as you define them, it doesn't matter.

Same applies to set theory, there's many, many set theories, and they differ between properties, and this has never caused problems, because in mathematical discussion or literature...you provide or point to a definition. So you'll never discuss about "set theory" without providing one or pointing to one.

This is very, very different from how normal people waste their time.

E.g. arguing whether AI "thinks" or not, but never defining what thinking means, thus you can't even conclude whether you think or not, because it's never been defined.