Definition:Natural Numbers/Zermelo Construction

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


 * $0 := \varnothing = \left\{{}\right\}$
 * $1 := \left\{{0}\right\} = \left\{{\varnothing}\right\}$
 * $2 := \left\{{1}\right\} = \left\{{\left\{{\varnothing}\right\}}\right\}$
 * $3 := \left\{{2}\right\} = \left\{{\left\{{\left\{{\varnothing}\right\}}\right\}}\right\}$
 * $\vdots$

Also see

 * Natural Numbers as Successor Sets for a more usual technique of axiomatization of $\N$.