# Definition:Natural Numbers/Von Neumann Construction/Successor Mapping

## Definition

Let $\omega$ denote the minimally inductive set.

Let the natural numbers $\N$ be modelled using the Von Neumann construction as the elements of $\omega$:

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

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

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

is the successor mapping on $\N$.