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 $\varnothing$ and successor sets, we thus define:


 * $0 := \varnothing = \left\{{}\right\}$
 * $1 := 0^+ = 0 \cup \left\{{0}\right\} = \left\{{0}\right\}$
 * $2 := 1^+ = 1 \cup \left\{{1}\right\} = \left\{{0, 1}\right\}$
 * $3 := 2^+ = 2 \cup \left\{{2}\right\} = \left\{{0, 1, 2}\right\}$
 * $\vdots$

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

Also see

 * Definition:Minimal Infinite Successor Set


 * Minimal Infinite Successor Set forms Peano Structure