# 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 the Empty Set: 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 Union: 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.

## Sources

- 1972: Patrick Suppes:
*Axiomatic Set Theory*(2nd ed.) ... (previous) ... (next): Preface to the First Edition