# 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.

## Source of Name

This entry was named for Ernst Friedrich Ferdinand Zermelo.

## Historical Note

The Zermelo construction of natural numbers was devised by Ernst Friedrich Ferdinand Zermelo.

While the approach is simple, it does not generalize easily to transfinite ordinals.

As a result, it is generally considered inferior to the von Neumann construction, which has considerable advantages over it.

Hence this approach is rarely seen, and noted for historical reasons only.