Definition:ZF

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.

Source of Name

This entry was named for Ernst Zermelo and Abraham Fraenkel.