> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/200-set-theory/set-theories/axiomatic-set-theory.md).

# Axiomatic set theory

<https://en.wikipedia.org/wiki/Axiomatic_set_theory>

As opposed to *naive set theory*, where the *axiom of comprehension* allows any property to constitute a set, **axiomatic set theory** is a system that restricts the sets which are allowed to be generated by strictly constraining them with the axioms.

The best known systems of axiomatic set theory are ZF (Zermelo-Fraenkel set theory) ZFC (ZF plus the axiom of choice), NBG (von Neumann-Bernays-Gödel set theory), MK (Morse-Kelley set theory), TG (RTarski-Grothendieck set theory).

## "Naive Set Theory" by Paul R. Halmos

1. Axiom of extension   &#x20;

   two sets are equal iff they have the same elements.
2. Axiom of unions   &#x20;

   for every collection of sets there exists a set that contains all the elements that belong to at least one set of the given collection.
3. Axiom of specification   &#x20;

   To every set A and to every condition S(x) there corresponds a set B whose elements are exactly those elements x of A for which S(x) holds.
4. Axiom of pairing   &#x20;

   For any two sets there exists a set that they both belong to.
5. Axiom of powers   &#x20;

   mFor each set there exists a collection of sets that contains among its elements all the subsets of the given set.
6. Axiom of infinity   &#x20;

   There exists a set containing 0 and the successor of each of its elements.
7. Axiom of substitution   &#x20;

   If S(a,b) is a sentence such that for each a in set A the set {b: S(a,b)} can be formed, then there exists a function F with domain A such that F(a) = {b:S(a,b)} for each a in A (Anything intelligent that one can do to the elements of a set yields a set).
8. Axiom of choice   &#x20;

   The Cartesian product of a non-empty family of non-empty sets is non-empty