Axiomatic set theory
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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).
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