Zermelo-Fraenkel Axioms
It would be very convenient indeed for set theorists if any collection of objects with a given property describable by the language of set theory could be called a set. Unfortunately, as shown by paradoxes such as Russell's Paradox, we must put some restrictions on which collections to call sets. The Zermelo Fraenkel axiom system, developed by Ernst Zermelo and Abraham Fraenkel, does precisely this.
Contents
The Axiom of Extensionability
This axiom establishes the most basic property of sets - a set is completely characterized by its elements alone.
Statement: If two sets have the same elements, they are identical
The Null Set Axiom
This axiom ensures that there is at least one set.
Statement: There exists a set called the null set which contains no elements.
The Axiom of Subset Selection
This axiom declares subsets of a given set as sets themselves.
Statement: Given a set 
, and a formula 
with one free variable, there exists a set whose elements are precisely those elements of which satisfy 
.
The Power Set Axiom
This axiom allows us to construct a bigger set from a given set.
Statement: Given a set , there is a set containing all the subsets of A and no other element.
The Axiom of Replacement
This axiom allows us, given a set, to construct other sets of the same size.
Statement: Given a set and a bijective binary relation describable in the language of set theory, there is a set which consists of exactly those elements related to elements in .
The Axiom of Union
This axiom allows us to take unions of two or more sets.
Statement: Given a set , there exists a set with exactly those elements which belong to some element of .
The Axiom of Infinity
This gives us at least one infinite set.
Statement: There exists a set containing the null set, such that for all 
in , 
is also in .
The Axiom of Foundation
This makes sure no set contains itself, thus avoiding certain paradoxical situations.
Statement: The relation belongs to is well-founded.
The Axiom of Choice
This allows to find a choice set for any arbitrary collection of sets.
Statement: Given any collection of sets, there exists a set (called the choice set) containing precisely one element of each set in the collection.
There is quite a bit of controversy over this axiom, because it does not hold for infinite sets. Generally, most mathematicians are uncomfortable using it, though it is widely accepted as being used for finite sets.
