Set Theory: In mathematics and logic: The theory of sets i.e. of "any gathering into a whole ... of distinct perceptual or mental objecs" or "A set is a many which allows itself to be thought of as one".
See first Set for a clarification of sets. There are quite a few axiomatizations of set theory. The general purpose of Set Theory is to have a set of assumptions about sets that is sufficient to serve as a foundation for all or most of mathematics.
The one that follows is the standard one known as ZF or ZermeloFrankel, after the men who thought of it. It consists of the following set of axioms, that are here given both by name and formally. Both the names and the axioms are standard:
ZF1: Axiom of Extensionality:
Two sets are equal precisely if they have the same elements.
x_{1}=x_{2} IFF (x_{3})(x_{3} e x_{1} IFF x_{3} e x_{2})
ZF2: Null Set Axiom:
There is a set without elements.
(Ex_{1})(x_{2})~(x_{2} e x_{1})
ZF3: Axiom of Pairing:
For any two sets there is a set which has precisely these
two sets as elements.
(x_{1})(x_{2})(Ex_{3})(x_{4})(x_{4} e x_{3} IFF x_{4}=x_{1} V x_{4}=x_{2})
ZF4: Axiom of Unions:
For any set there is a set with all the members of
all the members of the set.
(x_{1})(Ex_{2})(x_{3})(x_{3} e x_{2} IFF (Ex_{4})(x_{4} e x_{1} & x_{3} e x_{4}))
Def: Definition of setinclusion:
One set is included in another if all the elements of the one
are elements of the other.
x_{1} inc x_{2} IFF (x_{3})(x_{3} e x_{1} > x_{3} e x_{2})
ZF5: Power Set Axiom:
For every set there is a set which has as elements
all the sets that are included in the set.
(x_{1})(Ex_{2})(x_{3})(x_{3} e x_{2} IFF x_{3} inc x_{1})
ZF6: Axiom Schema of Replacement:
For every set there is a set such that if the first set
is by F related to the second set then for every set A
there is a set B such that every set C is element of A precisely if
C is related by F to B.
(x_{1})(Ex_{2})(F(x_{1},x_{2}) > (x_{3})(Ex_{4})(x_{5})(x_{5} e x_{4} IFF (Ex_{6})(x_{6} e x_{3} & F(x_{6},x_{5})))
Def: Definition of Null Set:
Ø is a term for the set without any elements.
x_{1}=Ø IFF ~(Ex_{2})(x_{2} e x_{1})
Def: Definition of Union:
x_{1} = (x_{2} U x_{3}) IFF (x_{4})(x_{4} e x_{1} IFF x_{4} e x_{2} V x_{4} e x_{3})
ZF7: Axiom of Infinity:
There is a set which has the Null Set as an element and
is such that union of the set and any element of the set also
is an element of the set.
(Ex_{1})(Ø e x_{1} & (x_{2})(x_{2} e x_{1} > (x_{1} U {x_{2}}) e x_{1})
ZF8: Axiom of Foundation:
Every nonempty set has an element
which has no element in common with the set.
(x_{1})(~(x_{1}=Ø) > (Ex_{2})(x_{2} e x_{1} & ~(Ex_{3})(x_{3} e x_{2} & x_{3} e x_{1}))
Here are some comments and explanations.
The Axiom Schema of Replacement is said to be a schema because it involves an arbitrary (schematic) function F.
Ø is a unique set by the Axiom of Extensionality and it exists by the Null Set Axiom.
The Axiom of Infinity implies that as Ø is element of the set, so are {Ø}, {{Ø}}, {{{Ø}}} ... etc. without end. These are all distinct since ~( Ø = {Ø}) etc. because Ø has no elements and {Ø} has one element, namely Ø.
The Axiom of Foundation prevents chains of the form (x _{1} e x_{2} & x_{2} e x_{3} & ... & x_{n1} e x_{1}). I.o.w.: it assures that no set is element of itself nor element of any of its elements.)
For a brief explanation of what is the point of it all  all of mathematics in one simple set of axioms and a lingua franca for all of mathematics  see Set.
There are many good introductions to Set Theory (and also some not so good). One of the best for the beginner must be Naive Set Theory by Paul Halmos. Those who want a somewhat stricter and fairly complete version of ZF should look into Suppes. Quine provides a sort of minimalist general set theory. Mulder is a recent survey of many set theories, their uses and foundations, and their relations to the foundations of physics and mathematics. Lipschutz is a fine and very clear survey and introduction.
