Definition:Natural Numbers/Zermelo Construction

Theorem
The natural numbers $\N = \set {0, 1, 2, 3, \ldots}$ can be defined as a series of subsets:


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

Thus the natural number $n$ consists of $\O$ enclosed in $n$ pairs of braces.

Also see

 * Definition:Natural Numbers for more usual techniques of defining $\N$.