Definition:Natural Numbers/Construction

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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.