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.

Symbolically:

$\forall Y: \map {\text{Fnc}} Y \implies \forall x: \exists y: \forall u: u \in y \iff \exists v: \tuple {v, u} \in Y \land v \in x$

where:

$\map {\text{Fnc}} X := \forall x, y, z: \tuple {x, y} \in X \land \tuple {x, z} \in X \implies y = z$

and the notation $\tuple {\cdot, \cdot}$ is understood to represent Kuratowski's formalization of ordered pairs.

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
• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $6$: Order Isomorphism and Transfinite Recursion: $\S 3$ The axiom of substitution: Definition $3.1$