# Definition:Natural Numbers/Von Neumann Construction

## Definition

Let $\omega$ denote the minimally inductive set.

The natural numbers can be defined as the elements of $\omega$.

Following Definition 2 of $\omega$, this amounts to defining the natural numbers as the finite ordinals.

In terms of the empty set $\O$ and successor sets, we thus define:

 $\ds 0$ $:=$ $\ds \O = \set {}$ $\ds 1$ $:=$ $\ds 0^+ = 0 \cup \set 0 = \set 0$ $\ds 2$ $:=$ $\ds 1^+ = 1 \cup \set 1 = \set {0, 1}$ $\ds 3$ $:=$ $\ds 2^+ = 2 \cup \set 2 = \set {0, 1, 2}$ $\ds$ $\vdots$ $\ds$ $\ds n + 1$ $:=$ $\ds n^+ = n \cup \set n$

This can be expressed in detail as:

 $\ds 0$ $:=$ $\ds \O = \set {}$ $\ds 1$ $:=$ $\ds \set \O$ $\ds 2$ $:=$ $\ds \set {\O, \set \O}$ $\ds 3$ $:=$ $\ds \set {\O, \set \O, \set {\O, \set \O} }$ $\ds$ $\vdots$ $\ds$

### Successor Mapping

The mapping $s: \N \to \N$ defined thus as:

$\forall n \in \N: \map s n = n + 1$

is the successor mapping on $\N$.

## Source of Name

This entry was named for John von Neumann.