Definition:Natural Numbers/Zermelo Construction

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

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

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

Also see

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.