# Definition:Natural Numbers/Construction

## Contents

## 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.