It is common to use foundations of mathematics in which ‘set’ is an undefined term; this is set theory as a foundation. In a pure material set theory like ZFC, every object is a set. Even in a structural approach such as ETCS, it is common for every object to be a structured set in some way or another.
Even if sets are not at the bottom of your foundations, still they are probably close. For instance, in type theory sometimes sets are defined as predicates on types, or as setoids. In homotopy type theory, sets are defined as 0-truncated homotopy types; see below.
Material set theory conflates two notions of sets, which were elegantly (but not first) described by Mathieu Dupont in a blog post as ‘set¹’ and ‘set²’, which we will here call ‘abstract set’ and ‘concrete set’. In the latter case (set²), we have some fixed universe of discourse (say, consisting of all real numbers), and a concrete set is a set of elements of this universe (so a set of real numbers, such as , , or ). In the former case (set¹), an abstract set is a purely abstract concept, an unstructured (except perhaps for the equality relation) collection of unlabelled elements. Arguably (this argument goes back at least to Lawvere), Cantor's original concept of cardinal number (Kardinalzahl) was the abstraction from a concrete set to its underlying abstract set.
Here we adopt a structural approach, in which a set is an abstract set, a (mostly) unstructured collection. For concrete sets, see subset; if the fixed universe of discourse is an abstract set , then a concrete set will be a subset of . In material set theory (as usually conceived), the fixed universe of discourse is an abstract set (or proper class) , an abstract set (internal to the theory) is an element of , and a concrete set is a sufficiently small subset of ; the concepts may be identified because comes with an isomorphism to its class of small subsets.
We still need to clarify exactly what sort of collection a set is. Although most foundations leave this unspecified, we may (perhaps circularly, and perhaps controversially) define it in various ways, as follows:
Each of these three adjectives describes a different aspect of what makes something a ‘set’, and they serve different purposes, which should not be conflated.
That a set is (in the category-theoretic sense) discrete is the most important point; nobody would call a category a ‘set’ merely because it is small and skeletal. (This clause is also the reason why it makes no difference whether we start from categories, -groupois, or -categories; discreteness immediately limits us to -groupoids.) The discreteness of a set reflects its lack of internal structure; while a category may have much structure relating its objects, the only structure remaining in a discrete category is whether any two given objects are isomorphic (and the fact that isomorphism is an equivalence relation); if (and only if) they are, then they are considered equal as elements of the set.
That a set is small allows there to be a collection of ‘all’ sets; this collection is not itself small, but we still have a large category of all sets, Set. A category that is merely discrete and skeletal may be called a class instead. But notice that the difference between a set and a class is a rather technical one; a class is just like a set, only (possibly) larger. Indeed, not everyone would agree with the requirement that a set must be small; although that is probably the most common way of talking to deal with size issues, it is not the only way. Another way, used by no less an authority than Categories Work, is to posit the existence of a strongly inaccessible cardinal and define a small set to be a set whose cardinality is less than .
That a set is skeletal is arguably the least important requirement; in fact, it is ‘evil’ in the technical sense of violating the principle of equivalence. A category that is merely small and discrete may be called a setoid instead. However, if you forbid yourself from referring to equality of elements of the setoid (which are the objects of the small discrete category) — or, equivalently, interpret “equality” to mean “isomorphism” therein — then you cannot distinguish the setoid from a set; each is merely a (small) collection of elements with an equivalence relation (called ‘equivalence’ in the setoid and ‘equality’ in the set, in both cases corresponding to isomorphism in a small discrete category). While size issues are real and cannot be ignored completely, it is possible to adopt foundations in which the issue of skeletality simply does not appear.
Definition is_set A := forall (x y : A) (p q : x == y), p == q
(There are numerous other equivalent definitions; see h-set.)
Regarding types as ∞-groupoids, this definition takes care only of the discreteness requirement. Regarding smallness, HoTT is usually formulated in Martin-Löf type theory with universes (type of types). If we have chosen one particular universe as the universe of “small” things, then “setness” could be restricted to types belonging to that universe.
Finally, skeletality is mostly meaningless in HoTT because paths are the only notion of (propositional) equality which we have to reason with. (There is also the notion of “definitional” equality, but as we cannot assert or prove statements of the form “ and are definitionally equal” inside the theory, this is mostly not relevant for the purposes of setness.)
|homotopy level||n-truncation||homotopy theory||higher category theory||higher topos theory||homotopy type theory|
|h-level 0||(-2)-truncated||contractible space||(-2)-groupoid||true/unit type/contractible type|
|h-level 1||(-1)-truncated||contractible-if-inhabited||(-1)-groupoid/truth value||(0,1)-sheaf/ideal||mere proposition/h-proposition|
|h-level 2||0-truncated||homotopy 0-type||0-groupoid/set||sheaf||h-set|
|h-level 3||1-truncated||homotopy 1-type||1-groupoid/groupoid||(2,1)-sheaf/stack||h-groupoid|
|h-level 4||2-truncated||homotopy 2-type||2-groupoid||(3,1)-sheaf/2-stack||h-2-groupoid|
|h-level 5||3-truncated||homotopy 3-type||3-groupoid||(4,1)-sheaf/3-stack||h-3-groupoid|
An early definition of “set” appears in
where in section 4 it says in translation
There are wholes which, although they contain the same parts A, B, C, D, …, nevertheless present themselves as different when seen from our point of view or conception (this kind of difference we call ‘essential’), e.g. a complete and a broken glass viewed as a drinking vessel.  A whole whose basic conception renders the arrangement of its parts a matter of indifference (and whose rearrangement therefore changes nothing essential from our point of view, if only that changes), I call a set.