Category:Axioms/Zermelo Universe Axioms

This category contains axioms related to Zermelo Universe Axioms.

A Zermelo universe is a basic universe which also satisfies the axiom of infinity:

$\text A 1$: 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.