Wikipedia 10K Redux by Reagle from Starling archive. Bugs abound!!!

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SetTheory

Sets are one of the base concepts of mathematics.  A SeT is, more or less, just a collection of objects, called its elements.  Standard notation uses braces around the list of elements, as in:

   {red, green, blue}
   {x : x is a primary color}

If every x in some set A is contained in some set B, then A is said to be a SubSet of B.  Every set has as subsets itself, called the improper subset, and the empty set {}.  The union of a collection of sets '''S''' = {S1,S2,S3,...} is the set of all elements contained in at least one of the sets S1,S2,S3,....The intersection of a collection of sets '''T''' = {T1,T2,T3,...} is the set of all elements contained in at least one of the sets T1,T2,T3,.... The union and intersection of sets, say A1,A2,A3,... are denoted

   A1 u A2 u A3 ...
   A1 n A2 n A3 ...

respectively.  If you don't mind jumping ahead a bit, the subsets of a given set form a BooleanAlgebra under these operations.  The set of all subsets of X is called its PowerSet and is denoted 2^X.

'''Axioms for set theory'''

The most common axioms for set theory are those of Zermelo and Fraenkel, which do a good job handling most sets '''they don't even handle the first set used as an example above, or the set used as an example in SeTs''' , and are equivalent to several other axiomatizations.  However, they aren't the only possible axiom schema '''they aren't an axiom schema at all.  They contain one axiom schema, the axiom of replacement''', and in some senses are to restrictive.  For instance, they provide a way to go from any set X to a set {X} and thereby prevent the existence of a set of all sets, which some other models allow.

The axioms of ZF are as follows:
   Axiom of extension:        Two sets are the same iff they have the same elements.
   Axiom of null set:         There is an empty set {}.
   Axiom of unordered pairs:  If X,Y are sets, so is {X,Y}.
   Axiom of unions:           For any set X, there is another set composed of the union of its elements.
   Axiom of infinite:         There exists a set X such that, if x is in X, so is {x}.
   Axiom of replacement:      Given any set and any MappinG, formally defined as a proposition P(x,y) where
                              P(x,y) and P(x,z) implies y=z, there is a set containing precisely the images of 
                              its elements.
   Axiom of PowerSet''s'':    There exists a set P(X) containing precisely the subsets of any given set X.
   Axiom of regularity:       Every non-empty set X contains some element Y such that X and Y are disjoint.

The AxiomOfChoice leads to the existence of several sets, like a well-ordering for the real numbers, that can't be constructed in ordinary ZF set theory.  However, it is impossible to prove they don't exist - the axiom is independent of the axioms of ZF - and so we are free to add it to our axioms if we want.  This has become more or less conventional, since most of the sets it adds are handy to have, even if we can never find an explicit representation for any of them.