# Definition:Natural Numbers/Von Neumann Construction

## Definition

Let $\omega$ denote the minimal infinite successor 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:

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

This can be expressed in detail as:

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

## Source of Name

This entry was named for John von Neumann.