# Contents

## 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 $x \in z$.