Ernst Zermelo and Abraham Fraenkel proposed the following axioms as a for what is now called Zermelo-Fraenkel set theory, or ZF. If this set of axioms are accepted along with the Axiom of Choice, it is often denoted ZFC.
Equality of sets: If and are sets, and iff , then .
Pair set: If and are sets, then there is a set containing only and .
Union (http://planetmath.org/Union) over a set: If is a set, then there exists a set that contains every element of each .
: If is a set, then there exists a set with the property that iff any element is also in .
Replacement axiom: Let be some formula. If, for all , there is exactly one such that is true, then for any set there exists a set with the property that iff there exists some such that is true.
: Let be some formula. If there is some that makes true, then there is a set such that is true, but for no is true.
: If is a set and is a condition on sets, there exists a set whose members are precisely the members of satisfying . (This axiom is also occasionally referred to as the ).
|Date of creation||2013-03-22 11:47:51|
|Last modified on||2013-03-22 11:47:51|
|Last modified by||mathcam (2727)|
|Synonym||Zermelo-Fraenkel set theory|