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
Axiom of extension
two sets are equal iff they have the same elements.
Axiom of unions
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.
Axiom of specification
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.
Axiom of pairing
For any two sets there exists a set that they both belong to.
Axiom of powers
mFor each set there exists a collection of sets that contains among its elements all the subsets of the given set.
Axiom of infinity
There exists a set containing 0 and the successor of each of its elements.
Axiom of substitution
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).
Axiom of choice
The Cartesian product of a non-empty family of non-empty sets is non-empty
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