# Definition:Natural Numbers/Zermelo Construction

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## 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