Definition:Zermelo-Fraenkel Set Theory

Definition
ZF is an abbreviation for Zermelo-Fraenkel Set Theory. It is a system of axiomatic set theory upon which the whole of (at least conventional) mathematics can be based.

Its basis consists of a system of Aristotelian logic, appropriately axiomatised, together with the Zermelo-Fraenkel axioms of Set Theory.

These are as follows:


 * The Axiom of Extension: Two sets are equal if and only if they have the same contents.


 * The Axiom of Existence: There exists a set that has no elements.


 * The Axiom of Pairing: For any two sets, there exists a set to which only those two sets belong.


 * The Axiom of Subsets: For every set and every condition, there corresponds a set whose elements are exactly the same as those elements of the original set for which the condition is true.


 * The Axiom of Unions: For every collection of sets, there exists a set that contains all the elements that belong to at least one of the sets in the collection.


 * The Axiom of Powers: For each set, there exists a collection of sets that contains amongst its elements all the subsets of the given set.


 * The Axiom of Infinity: There exists a set containing a set with no elements and the successor of each of its elements.


 * The Axiom of Replacement: For any set $S$, there exists a set $x$ such that, for any element $y$ of $S$, if there exists an element $z$ satisfying the condition $P \left({y, z}\right)$ (where $P \left({y, z}\right)$ is a propositional function), then such $z$ appear in $x$.


 * The Axiom of Foundation: For all non-null sets, there is an element of the set that shares no member with the set.

Note that in this system, the (controversial) Axiom of Choice is not included.