Definition:Zermelo-Fraenkel Set Theory

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: Let $$P \left({y, z}\right)$$ be a propositional function, which determines a function.

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.