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 $\N$.

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.

Sources

• 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 1$ Preliminaries