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The '''Zermelo-Fraenkel axioms''' form one of several possible formulations of [[Set_theory#Axiomatic_set_theory|axiomatic set theory]].

==The axioms==
There are eight Zermelo-Fraenkel (ZF) axioms:<ref name=Jech>

{{cite book |title=Set theory |author=Thomas J Jech |url= http://books.google.com/books?id=pLxq0myANiEC&pg=PA1#v=onepage&q&f=false |isbn=0123819504 |year=1978 |publisher=Academic Press}}

</ref>
#<u>Axiom of extensionality</u>: If ''X'' and ''Y'' have the same elements, then ''X=Y''
#<u>Axiom of pairing</u>: For any ''a'' and ''b'' there exists a set {''a, b''} that contains exactly ''a'' and ''b''
#<u>Axiom schema of separation</u>: If &phi; is a property with parameter ''p'', then for any ''X'' and ''p'' there exists a set ''Y'' that contains all those elements ''u''&isin;''X'' that have the property &phi;; that is, the set ''Y''={''u''&isin;''X'' | &phi;''(u, p)''}
#<u>Axiom of union</u>: For any set ''X'' there exists a set {{nowrap|''Y'' <nowiki>=</nowiki> <big><big>&cup;</big></big>''X'',}} the union of all elements of ''X''
#<u>Axiom of power set</u>: For any ''X'' there exists a set ''Y''=''P(X)'', the set of all subsets of ''X''
#<u>Axiom of infinity</u>: There exists an infinite set
#<u>Axiom schema of replacement</u>: If ''f'' is a function, then for any ''X'' there exists a set ''Y'', denoted ''F(X)'' such that ''F(X)''={''f(x)''|''x''&isin;''X''}
#<u>Axiom of regularity</u>: Every nonempty set has an &isin;-minimal element

If to these is added the axiom of choice, the theory is designated as the ZFC theory:

&emsp;9. <u>Axiom of choice</u>: Every family of nonempty sets has a choice function

For further discussion of these axioms, see the [http://en.citizendium.org/wiki/Zermelo-Fraenkel_axioms/Bibliography bibliography] and the [[Zermelo-Fraenkel_axioms/External_Links|linked articles]].

==References==
<references/>