Definition:Zermelo-Fraenkel Universe

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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.