Aleph Zero equals Cardinality of Naturals
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Theorem
$\aleph_0 = \card \N$
where
- $\aleph$ denotes the aleph mapping,
- $\card \N$ denotes the cardinality of $\N$.
Proof
| \(\ds \aleph_0\) | \(=\) | \(\ds \card {\aleph_0}\) | Cardinal of Cardinal Equal to Cardinal | |||||||||||
| \(\ds \) | \(=\) | \(\ds \card {\omega}\) | Definition of Aleph Mapping | |||||||||||
| \(\ds \) | \(=\) | \(\ds \card {\N}\) | Definition of Natural Numbers |
$\blacksquare$
Sources
- Mizar article CARD_1:47