Is there an axiomatisation of some kind of category theory and a definition of sets in this framework such that the axioms of ZF resp. ZFC are theorems?
[Math] ZF(C) and category theory
ct.category-theoryfoundationsreference-requestset-theory
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On the subject of categorical versus set-theoretic foundations there is too much complicated discussion about structure that misses the essential point about whether "collections" are necessary.
It doesn't matter exactly what your personal list of mathematical requirements may be -- rings, the category of them, fibrations, 2-categories or whatever -- developing the appropriate foundational system for it is just a matter of "programming", once you understand the general setting.
The crucial issue is whether you are taken in by the Great Set-Theoretic Swindle that mathematics depends on collections (completed infinities). (I am sorry that it is necessary to use strong language here in order to flag the fact that I reject a widely held but mistaken opinion.)
Set theory as a purported foundation for mathematics does not and cannot turn collections into objects. It just axiomatises some of the intuitions about how we would like to handle collections, based on the relationship called "inhabits" (eg "Paul inhabits London", "3 inhabits N"). This binary relation, written $\epsilon$, is formalised using first order predicate calculus, usually with just one sort, the universe of sets. The familiar axioms of (whichever) set theory are formulae in first order predicate calculus together with $\epsilon$.
(There are better and more modern ways of capturing the intuitions about collections, based on the whole of the 20th century's experience of algebra and other subjects, for example using pretoposes and arithmetic universes, but they would be a technical distraction from the main foundational issue.)
Lawvere's "Elementary Theory of the Category of Sets" axiomatises some of the intuitions about the category of sets, using the same methodology. Now there are two sorts (the members of one are called "objects" or "sets" and of the other "morphisms" or "functions"). The axioms of a category or of an elementary topos are formulae in first order predicate calculus together with domain, codomain, identity and composition.
Set theorists claim that this use of category theory for foundations depends on prior use of set theory, on the grounds that you need to start with "the collection of objects" and "the collection of morphisms". Curiously, they think that their own approach is immune to the same criticism.
I would like to make it clear that I do NOT share this view of Lawvere's.
Prior to 1870 completed infinities were considered to be nonsense.
When you learned arithmetic at primary school, you learned some rules that said that, when you had certain symbols on the page in front of you, such as "5+7", you could add certain other symbols, in this case "=12". If you followed the rules correctly, the teacher gave you a gold star, but if you broke them you were told off.
Maybe you learned another set of rules about how you could add lines and circles to a geometrical figure ("Euclidean geometry"). Or another one involving "integration by parts". And so on. NEVER was there a "completed infinity".
Whilst the mainstream of pure mathematics allowed itself to be seduced by completed infinities in set theory, symbolic logic continued and continues to formulate systems of rules that permit certain additions to be made to arrays of characters written on a page. There are many different systems -- the point of my opening paragraph is that you can design your own system to meet your own mathematical requirements -- but a certain degree of uniformity has been achieved in the way that they are presented.
We need an inexhaustible supply of VARIABLES for which we may substitute.
There are FUNCTION SYMBOLS that form terms from variables and other terms.
There are BASE TYPES such as 0 and N, and CONSTRUCTORS for forming new types, such as $\times$, $+$, $/$, $\to$, ....
There are TRUTH VALUES ($\bot$ and $\top$), RELATION SYMBOLS ($=$) and CONNECTIVES and QUANTIFIERS for forming new predicates.
Each variable has a type, formation of terms and predicates must respect certain typing rules, and each formation, equality or assertion of a predicate is made in the CONTEXT of certain type-assignments and assumptions.
There are RULES for asserting equations, predicates, etc.
We can, for example, formulate ZERMELO TYPE THEORY in this style. It has type-constructors called powerset and {x:X|p(x)} and a relation-symbol called $\epsilon$. Obviously I am not going to write out all of the details here, but it is not difficult to make this agree with what ordinary mathematicians call "set theory" and is adequate for most of their requirements
Alternatively, one can formulate the theory of an elementary topos is this style, or any other categorical structure that you require. Then a "ring" is a type together with some morphisms for which certain equations are provable.
If you want to talk about "the category of sets" or "the category of rings" WITHIN your tpe theory then this can be done by adding types known as "universes", terms that give names to objects in the internal category of sets and a dependent type that provides a way of externalising the internal sets.
So, although the methodology is the one that is practised by type theorists, it can equally well be used for category theory and the traditional purposes of pure mathematics. (In fact, it is better to formalise a type theory such as my "Zermelo type theory" and then use a uniform construction to turn it into a category such as a topos. This is easier because the associativity of composition is awkward to handle in a recursive setting. However, this is a technical footnote.)
A lot of these ideas are covered in my book "Practical Foundations of Mathematics" (CUP 1999), http://www.PaulTaylor.EU/Practical-Foundations Since writing the book I have written things in a more type-theoretic than categorical style, but they are equivalent. My programme called "Abstract Stone Duality", http://www.PaulTaylor.EU/ASD is an example of the methodology above, but far more radical than the context of this question in its rejection of set theory, ie I see toposes as being just as bad.
The main fact is that a very weak meta-theory typically suffices, for theorems about models of set theory. Indeed, for almost all of the meta-mathematical results in set theory with which I am familiar, using only PA or considerably less in the meta-theory is more than sufficient.
Consider a typical forcing argument. Even though set theorists consider ZFC plus large cardinals, supercompact cardinals and extendible cardinals and more — very strong object theories — they need very little in the meta-theory to undertake the relative consistency proofs they have in mind. To show for example that the consistency of ZFC plus a supercompact cardinal implies the consistency of ZFC plus PFA, there is a forcing argument involved, the Baumgartner forcing. But the meta-theory does not need to undertake the forcing itself, but only to prove that forcing over an already-given model of ZFC plus a supercompact cardinal works as described. And that can be proved in a very weak theory such as PA or even much weaker.
So we don't really even use set theory in the meta-theory but just some weak arithmetic theory. I expect that the proof theorists can likely tell you much weaker theories than PA that suffice for the meta-theory of most set-theoretic meta-mathematical arguments.
Best Answer
Yes, this is a well-known fact that goes back to Cole, Mitchell, and Osius in the 70's. The relevant kind of category is, as Harry says in the comments, a well-pointed topos with extra properties; Lawvere's Elementary Theory of the Category of Sets is an axiomatization of such a category. To define ZF-sets in such a category you build them along with their hereditary membership structure as well-founded graphs; there is a good introduction in Chapter VI of Mac Lane & Moerdijk's book Sheaves in Geometry and Logic. If you want a very detailed treatment that includes analogous results for a wide variety of set theories including both ZF and ZFC and also weaker and constructive versions, there is my own paper Comparing material and structural set theories.