basic constructions:
strong axioms
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A set theory is a theory of sets.
Naïve set theory is the basic algebra of the subsets of any given set $U$, together with a few levels of power sets, say up to $\mathcal{P}\mathcal{P}\mathcal{P}U$ and possibly no further. Often students see this first for the set of real numbers as $U$ (although in fact one could start with the set of natural numbers and go one level further for an equivalent theory). One could also use function sets instead of power sets as the basic set-forming operation (especially for a weakly predicative theory in constructive mathematics).
Once you start thinking very much about the nature of sets in general (rather than merely using naïve set theory), it quickly becomes clear that one must be careful about how one can and cannot form sets. Georg Cantor is credited with being the first to think about sets this deeply; although he did not propose a system of general rules for valid set-making operations, he recognised that some sets were ‘inconsistent’. An axiomatic set theory is a set theory which carefully states the rules (or ‘axioms’) which sets are assumed to obey.
Gottlob Frege has perhaps the first axiomatic set theory, but it was found (by Bertrand Russell) to be logically trivial; see Russell's paradox. Ernst Zermelo is credited with the first consistent axiomatic set theory.
There are two ways to go about doing axiomatic set theory. The more ambitious is to develop a foundational set theory: set theory as foundations for all of mathematics. This is what Frege proposed (although he failed through inconsistency) and which Zermelo achieved.
A variation of Zermelo's system (developed by Fraenkel and Skolem and called Zermelo–Fraenkel set theory or ZFC) is the orthodox foundations today, although it needs to be supplemented by Grothendieck universes (or something along those lines) to handle modern category theory, and set theorists often consider further strengthenings of it through large cardinal axioms (of which the existence of Grothendieck universes is an example, just the tip of the iceberg).
It is also possible to make a definitional set theory, in which one defines sets in terms of some more primitive concept. For example, Bill Lawvere proposed a foundation ETCC based on the category of categories; then a set may be defined as a discrete category. This is also the case in homotopy type theory, where sets are defined as those types which are h-sets.
In constructive mathematics, a foundation based on type theory is popular, with types interpreted as presets (sets without equality); then a set may be defined as a preset equipped with an equivalence relation (the term setoid is also used for such a gadget). In computer science, a foundation based on the lambda-calculus is sometimes seen; in these terms, the concept of list is more natural than set, with the difference being that sets have a coarser notion of equality.
On the $n$Lab we like to distinguish between two types of set theory, especially in foundations:
In a material set theory (also called a membership-based set theory), the elements of a set exist independently of that set. As such, it makes sense to take two completely unrelated sets and ask if two of their members are equal, and sometimes the answer will be Yes. Frequently in material set theory one takes everything to be a pure set, so that elements of sets are themselves sets. Therefore, any two sets may be meaningfully compared to ask if they are equal or if one is a member of the other.
A structural set theory, on the other hand, looks more like type theory. Here, the elements of a set have no existence or structure apart from their identity as elements of that set. In particular, they are not themselves sets, and cannot be elements of any other set, at least not without invoking some explicit type-casting? operator (which here could simply be a function from one set to the other). Similarly, elements of different sets cannot be compared to each other (without type-casting them to become elements of the same set).
Among category theorists, it's popular to state the axioms of a structural set theory by specifying elementary properties of the category of sets; the orthodoxy here (to the extent that there is one) is probably Bill Lawvere's ETCS, which suffices for most everyday uses but must be supplemented to handle some esoteric parts of modern mathematics. Another structural set theory, which is stronger than ETCS and less closely tied to category theory, is SEAR.
In contrast, ZFC is an example of a material set theory. From a model of either kind of set theory we can construct a model of the other, so the two are, broadly speaking, equivalent; for example, SEARC (SEAR with the axiom of choice) is equivalent in this way to ZFC, while ETCS is equivalent to a weak (‘bounded’) variation of ZFC. A more precise statement is that the two kinds of theories form categories related by the material-structural adjunction.
Category theory can provide a common model theory to compare various set theories. Although only structural set theories like ETCS treat the elementary properties of the category $Set$ of sets as fundamental, one can ask for any set theory what properties $Set$ satisfies and compare them in those terms. At the very least, $Set$ should be a pretopos.
There is also algebraic set theory, in which a material set theory is interpreted in the internal logic of some ambient category, often called a “category of classes”. See also stack semantics.
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Formalization of set theory in homotopy type theory (via h-sets) is discussed in
Egbert Rijke, Bas Spitters, Sets in homotopy type theory (arXiv:1305.3835)
Univalent Foundations Project, section 3 of Homotopy Type Theory -- Univalent Foundations of Mathematics
Jérémy Ledent, Modeling set theory in homotopy type theory (2014) (LedentSetinHoTT.pdf?)