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 $3.1$