# Definition:Zermelo-Fraenkel Universe

## Definition

A Zermelo-Fraenkel universe is a basic universe which also satisfies the axiom of infinity and the axiom of replacement:

### $\text A 1$: Axiom of Transitivity

$V$ is a transitive class.

### $\text A 2$: Axiom of Swelledness

$V$ is a swelled class.

### $\text A 3$: Axiom of the Empty Set

The empty class $\O$ is a set, that is:

$\O \in V$

### $\text A 4$: Axiom of Pairing

Let $a$ and $b$ be sets.

Then the class $\set {a, b}$ is likewise a set.

### $\text A 5$: Axiom of Unions

Let $x$ be a set (of sets).

Then its union $\bigcup x$ is also a set.

### $\text A 6$: Axiom of Powers

Let $x$ be a set.

Then its power set $\powerset x$ is also a set.

### $\text A 7$: Axiom of Infinity

Let $\omega$ be the class of natural numbers as constructed by the Von Neumann construction:

 $\ds 0$ $:=$ $\ds \O$ $\ds 1$ $:=$ $\ds 0 \cup \set 0$ $\ds 2$ $:=$ $\ds 1 \cup \set 1$ $\ds 3$ $:=$ $\ds 2 \cup \set 2$ $\ds$ $\vdots$ $\ds$ $\ds n + 1$ $:=$ $\ds n \cup \set n$ $\ds$ $\vdots$ $\ds$

Then $\omega$ is a set.

### $\text A 8$: Axiom of Replacement

For every mapping $f$ and set $x$ in the domain of $f$, the image $f \sqbrk x$ is a set.

## Source of Name

This entry was named for Ernst Zermelo and Abraham Fraenkel.