Definition:Natural Numbers/Elements of Minimal Infinite Successor Set

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Definition

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:

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


This can be expressed in detail as:

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


Also see


Sources