Aleph Product is Aleph

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Theorem

Let $x$ be an ordinal.


Then:

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

where $\aleph$ denotes the aleph mapping.


Proof

\(\displaystyle \left\vert{\aleph_x \times \aleph_x }\right\vert\) \(=\) \(\displaystyle \left\vert{\aleph_x}\right\vert\) Non-Finite Cardinal is equal to Cardinal Product
\(\displaystyle \) \(=\) \(\displaystyle \aleph_x\) Aleph is Infinite Cardinal

$\blacksquare$


Sources