Set Theory/Zermelo‒Fraenkel set theory

Zermelo‒Fraenkel set theory Zermelo‒Fraenkel set theory (ZF set theory) is the theory given by the following axioms (where most are written in the language of set theory ): Axiom of extensionality: Empty set: Pairing: Union: Restricted comprehension schema: Given a set and a formula in the language of set theory . Then we have a set of all such that is true. We write this as the set . Replacement schema: Given a set and a formula in the language of set theory , such that for all there exists a unique such that is true. Then there exists a set whose elements are all those . We write this as the set . Power set: Infinity: There exists an inductive set. An inductive set is one that contains an empty set, and if it contains , then it contains . Note that in the Infinity axiom, we said that the inductive set contains `an' empty set. We will soon see that there is in fact a unique empty set, so we can change our definition of an inductive set to one that contains `the' empty set.