Non-Finite Cardinal is equal to Cardinal Product/Corollary

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Corollary to Non-Finite Cardinal is equal to Cardinal Product

Let $S$ be a set that is equinumerous to its cardinal number.

Let $\left|{ S }\right|$ denote the cardinal number of $S$.

Let:

$\left|{S}\right| \ge \omega$

where $\omega$ denotes the minimal infinite successor set.


Then:

$\left|{S \times S}\right| = \left|{S}\right|$


Proof

By the hypothesis:

$S \sim \left|{S}\right|$


By Cartesian Product Preserves Cardinality:

$S \times S \sim \left|{S}\right| \times \left|{S}\right|$

Therefore:

\(\displaystyle \left\vert{S \times S}\right\vert\) \(=\) \(\displaystyle \left\vert{\left\vert{S}\right\vert \times \left\vert{S}\right\vert}\right\vert\) Equivalent Sets have Equal Cardinal Numbers
\(\displaystyle \) \(=\) \(\displaystyle \left\vert{S}\right\vert\) Non-Finite Cardinal is equal to Cardinal Product

$\blacksquare$


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