# Definition:Zermelo-Fraenkel Universe

## Contents

## Definition

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

### $\text A 1$: 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 $\displaystyle \bigcup x$ 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:

\(\displaystyle 0\) | \(:=\) | \(\displaystyle \O\) | |||||||||||

\(\displaystyle 1\) | \(:=\) | \(\displaystyle 0 \cup \set 0\) | |||||||||||

\(\displaystyle 2\) | \(:=\) | \(\displaystyle 1 \cup \set 1\) | |||||||||||

\(\displaystyle 3\) | \(:=\) | \(\displaystyle 2 \cup \set 2\) | |||||||||||

\(\displaystyle \) | \(\vdots\) | \(\displaystyle \) | |||||||||||

\(\displaystyle n + 1\) | \(:=\) | \(\displaystyle n \cup \set n\) | |||||||||||

\(\displaystyle \) | \(\vdots\) | \(\displaystyle \) |

Then $\omega$ is a set.

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

For any function $f$ and subset $S$ of the domain of $f$, there is a set containing the image $\map f S$.

More formally, let us express this as follows:

Let $\map P {y, z}$ be a propositional function, which determines a function.

That is, we have:

- $\forall y: \exists x: \forall z: \paren {\map P {y, z} \iff x = z}$.

Then we state as an axiom:

- $\forall w: \exists x: \forall y: \paren {y \in w \implies \paren {\forall z: \paren {\map P {y, z} \implies z \in x} } }$

## Source of Name

This entry was named for Ernst Zermelo and Abraham Fraenkel.

## Sources

- 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 2$ Transitivity and supercompleteness