Definition:Natural Numbers/Natural Numbers from Nesting of Subsets

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Theorem

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

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


However, this approach is rarely seen, as it is less useful that the more prevalent method of construction as Elements of Minimal Infinite Successor Set.


Also see


Sources