Example 4.7.2.10 (The First Infinite Cardinal). We let $\aleph _0$ denote the smallest infinite cardinal. Alternatively, $\aleph _0$ can be defined as the ordinal $\omega $ of Example 4.7.1.10 (the order type of the linearly ordered set $\{ 0 < 1 < 2 < \cdots \} $). A set $S$ has cardinality $\aleph _0$ if and only if it is countably infinite.
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