Set is Countable iff Cardinality not greater Aleph Zero
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Theorem
Let $X$ be a set.
Then:
- $X$ is countable if and only if $\size X \le \aleph_0$
where
- $\size X$ denotes the cardinality of $X$,
- $\aleph$ denotes the aleph mapping.
Proof
- $X$ is countable
- there exists an injection $X \to \N$ by definition of countable set
- $\size X \le \size \N$ by Injection iff Cardinal Inequality
- $\size X \le \aleph_0$ by Aleph Zero equals Cardinality of Naturals.
$\blacksquare$