Aleph Product is Aleph

Theorem

Let $x$ be an ordinal.

Then:

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

where $\aleph$ denotes the aleph mapping.

Proof

\(\ds \left\vert{\aleph_x \times \aleph_x }\right\vert\)

\(=\)

\(\ds \left\vert{\aleph_x}\right\vert\)

Non-Finite Cardinal is equal to Cardinal Product

\(\ds \)

\(=\)

\(\ds \aleph_x\)

Aleph is Infinite Cardinal

$\blacksquare$

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 10.46 \ (2)$