Aleph Product is Aleph
Jump to navigation
Jump to search
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: $\S 10.46 \ (2)$