Minimal Infinite Successor Set is Infinite Cardinal
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Theorem
$\omega$, the minimal infinite successor set, is an element of the infinite cardinal class $\mathcal N’$.
Proof
By Cardinal Number Less than Ordinal: Corollary:
- $\left\vert{\omega}\right\vert \le \omega$
Moreover, for any $n \in \omega$, by Cardinal of Finite Ordinal:
- $\left\vert{n}\right\vert < \left\vert{n+1}\right\vert \le \left\vert{\omega}\right\vert$
Thus by Cardinal of Finite Ordinal:
- $n \in \left\vert{\omega}\right\vert$
Therefore:
- $\omega = \left\vert{\omega}\right\vert$
By Cardinal of Cardinal Equal to Cardinal: Corollary:
- $\omega \in \mathcal N’$
$\blacksquare$