Definition:Natural Numbers/Construction

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Definition

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


Elements of Minimal Infinite Successor Set

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:

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


Natural Numbers in Real Numbers

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

Let $\mathcal I$ be the set of all inductive sets in $\R$.


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

$\N := \displaystyle \bigcap \mathcal I$

where $\displaystyle \bigcap$ denotes intersection.


Natural Numbers from Nesting of Subsets

The natural numbers $\N = \left\{{0, 1, 2, 3, \ldots}\right\}$ can be defined as a series of subsets:

$0 := \varnothing = \left\{{}\right\}$
$1 := \left\{{0}\right\} = \left\{{\varnothing}\right\}$
$2 := \left\{{1}\right\} = \left\{{\left\{{\varnothing}\right\}}\right\}$
$3 := \left\{{2}\right\} = \left\{{\left\{{\left\{{\varnothing}\right\}}\right\}}\right\}$
$\vdots$