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 $\card S$ denote the cardinal number of $S$.
Let:
- $\card S \ge \omega$
where $\omega$ denotes the minimally inductive set.
Then:
- $\card {S \times S} = \card S$
Proof
By hypothesis:
- $S \sim \card S$
By Cartesian Product Preserves Cardinality:
- $S \times S \sim \card S \times \card S$
Therefore:
\(\ds \card {S \times S}\) | \(=\) | \(\ds \card {\card S \times \card S}\) | Equivalent Sets have Equal Cardinal Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \card S\) | Non-Finite Cardinal is equal to Cardinal Product |
$\blacksquare$
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 10.34$