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The "atomatic" forms of well formed formula are (a) x is a set, (b) x is in y, and (c) x is y, whose notation is "x=y". Any other new words or types of sentences have to be introduced by "Definition". The following are chosen sentences called "Axioms", which are the "initial setup" for deduction.
(Extensionality axiom) For any sets A and B, "for any set x, x is in A if and only if x is in B" if and only if A=B.
(Empty set axiom) There exists a set B such that for any set x, x is not in B.
(Intersection axiom) For any sets A and B, there exists a set C such that " for any set x, x is in C if and only if x is in A and in B ".
Note that in the following text, any variable with no notation of what it is, it will be all sets. For example, I will write "for any A, ...." and "there is a B such that ..." instead of "for any set A,..." and "there is a set B such that ....", respectively. We still have the following axioms:
(Union axiom) For any A, there exists a U such that " for any x, x is in U if and only if x is in A or in B ".
(Pairing axiom) For any u, v, there exists a C such that " for any x, x is in C if and only if x is u or is v ".
From these axioms we can build up the concepts of intersections AIB, unions AUB, of given A and B. Individually we have gotten the set --- the empty set o, and for any sets u, v, the set of u and v, denoted by {u,v}. However, we can intuitively find that we can take away some objects from a given set, but there is no axiom to support this behavior.
The following are some modified axiom:
(modified union axiom) For a set A, there is an T, such that for any x, x is in T if and only if "x is in B and B is in T, for some B".
(Axiom schema) For a set A, there is a set X such that for any x, "x is in X if and only if x is in A and P(x).
Note that the axiom is called "schema" because it offers a way for producing axioms whenever P(x) is filled in with any sentence.