How Do Mathematicians Interpret the Concept of Set Outside Pure Set Theory?

[Werg22]

I'd like to know how most mathematicians view the concept of set. When used in a context outside of set theory, does the word "set" take a meaning (as opposed to leaving it as an undefined term in a set theory) that does in fact follow the axioms of a complete set theory, and therefore all deductions (theorems) that are given to us by such a set theory are also valid for this meaningful set concept?

 

[matt grime]

A set is small. A proper class is large. Practically, for me, proper classes will come up when ever I need to have set indexed things, thus one will (usually) have a proper class of objects in your category. For example, set theory rarely intrudes into the world of finite dimensional representations of a group - there is a set of isomorphism classes of these things, though there is a proper class of fin dim reps. The category of fin dim reps is called skeletally small owing to this fact.

But if one wishes to allow arbitrary products and coproducts (i.e. indexed by arbitrary sets), then there is no way to get round the fact that you now have a proper class of pair-wise non-isomorphic objects.

 

[HallsofIvy]

matt, I don't believe that answers the question that was intended.

Werg22, in applications of set theory "outside" of set theory, specific sets are defined. The term "Set", itself, is "given" from set theory.