Note that "a working mathematician" is probably not the best choice of words, it's supposed to mean "someone who needs the theory for applications rather than for its own sake". Think about it as a homage to Mac Lane's classic. I'm in no way implying that set theory is not "real mathematics" (whatever that expression might mean, though I've heard some people say it, and I don't respect this point of view that something abstract is "not real mathematics") and I have a great respect for that field of study.

However, I'm personally not interested in set theory and its logic for their own sake (as of now). For a while I have treated them naively, and it was fine as I haven't needed anything beyond introductory chapters in compheresnive books on algebra, analysis or topology. But recently I decided to understand the foundations of category theory based on Grothendieck universes and inaccessible cardinals. So, I went to read some sources on set theory. And was really confused at first about such definitions as of a "transitive set", which implicitly assume that all elements of all sets are sets. Then I read more about it and discovered that in $\mathrm{ZFC}$

everything is a set!

It seemed absurd to me at first. After consulting several sources, I realized that ZFC was meant to be a (or even *the*) foundation for mathematics, rather than simply a theory which gives us a framework to work with sets, so at that time people thought that every mathematical object can be defined in term of sets. It didn't seem as unreasonable as before anymore, but still...

It still doesn't feel right for me. I understand that at the time when Zermelo and Fraenkel were developing axiomatic set theory, it was reasonable to think that every conceivable mathematical object is set. But it was a long time ago; is it still this way - especially concerning category theory?

If we work in $\mathrm{ZFC}$ (+ $\mathrm{UA}$) we have to assume that every object in any category is a actually as set. And the same should go for morphisms. Because, given a category $\mathrm{C}$, $\operatorname{ob} \mathrm{C}$ and $\operatorname{mor} \mathrm{C}$ are sets, so their elements, namely, objects and morphisms of $\mathrm{C}$, should also be sets.

**The question is**: is the assumption that there are no urelements, that is, that every conceivable mathematical object can be modeled in term of sets, reasonable, as of the second decade of the 21st century? Is there an area of mathematics where we need urelements? Can this way of thinking be a burden in some mathematical fields? (Actually, it's three questions, sorry. But they are related)

**P.S.** I hope this question is not too "elementary" for this site. But as I understand there are quite a lot of working mathematicians who don't think much about foundations. So, even if this question is not *useful* for them, it can at least be *interesting* for them.

Everything is a set$\endgroup$in some universe of set theory.8more comments