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:


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


 * $n + 1 := n^+ = n \cup \set n$

This can be expressed in detail as:
 * $0 := \O$
 * $1 := \set \O$
 * $2 := \set {\O, \set \O}$
 * $3 := \set {\O, \set \O, \set {\O, \set \O} }$
 * $\vdots$

Also see

 * Definition:Minimal Infinite Successor Set


 * Minimal Infinite Successor Set forms Peano Structure