# Definition:Natural Numbers/Construction

## Definition

The natural numbers $\N$ can be constructed in the following ways:

### Von Neumann Construction of Natural Numbers

Let $\omega$ denote the minimal infinite successor set.

The natural numbers can be defined as the elements of $\omega$.

Following Definition 2 of $\omega$, this amounts to defining the natural numbers as the finite ordinals.

In terms of the empty set $\O$ and successor sets, we thus define:

 $\displaystyle 0$ $:=$ $\displaystyle \O = \set {}$ $\displaystyle 1$ $:=$ $\displaystyle 0^+ = 0 \cup \set 0 = \set 0$ $\displaystyle 2$ $:=$ $\displaystyle 1^+ = 1 \cup \set 1 = \set {0, 1}$ $\displaystyle 3$ $:=$ $\displaystyle 2^+ = 2 \cup \set 2 = \set {0, 1, 2}$ $\displaystyle$ $\vdots$ $\displaystyle$ $\displaystyle n + 1$ $:=$ $\displaystyle n^+ = n \cup \set n$

### Inductive Set Definition for Natural Numbers

Let $x$ be a set which is an element of every inductive set.

Then $x$ is a natural number.

### Inductive Set Definition for Natural Numbers in Real Numbers

Let $\R$ be the set of real numbers.

Let $\II$ be the set of all inductive sets defined as subsets of $\R$.

Then the natural numbers $\N$ are defined as:

$\N := \displaystyle \bigcap \II$

where $\displaystyle \bigcap$ denotes intersection.

### Zermelo Construction of Natural Numbers

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.