# 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 |

$\blacksquare$

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 10.34$