nLab natural numbers structure

Context

Set theory

Idea
Definition
Foundational concerns
See also

Idea

Abstracting the von Neumann ordinals and the Zermelo ordinals, amongst other definitions of the natural numbers, in material set theory.

Definition

Given a set $V$ with an extensional relation $\prec$ , a natural numbers structure is an element $\mathbb{N} \in V$ , an element $0 \in V$ , and a function $s:V \to V$ such that

$0 \prec \mathbb{N}$
for all $x \in V$ , if $x \prec \mathbb{N}$ , then $s(x) \prec \mathbb{N}$
for all $z \in V$ , if $0 \prec z$ and for all $x \in V$ , if $x \prec z$ , then $s(x) \prec z$ , then for all $x \in V$ , $x \prec \mathbb{N}$ implies that $x \prec z$ .

Foundational concerns

Usually in material set theory, $0$ is defined to be the empty set $\emptyset$ , $\mathbb{N}$ is defined to be the von Neumann ordinal $\omega$ , and $s$ is defined to be the operation $x \mapsto x \cup \{x\}$ . However, there are alternative options, such as using the Zermelo ordinals, where the operation $s$ is given by $x \mapsto \{x\}$ , amongst others.

One could avoid having to choose amongst the definitions and abstract it all by adding primitive constants $\mathbb{N}$ and $0$ and primitive unary operation $s$ to the set theory, satisfying the axiom

$0 \in \mathbb{N}$ ; for all sets $x$ , if $x \in \mathbb{N}$ , then $s(x) \in \mathbb{N}$ ; and for all sets $z$ , if $0 \in z$ and for all sets $x$ , if $x \in z$ , then $s(x) \in z$ , then for all sets $x$ , $x \in \mathbb{N}$ implies that $x \in z$ .