Definition 4.7.2.1. Let $S$ be a set. We let $|S|$ denote the smallest ordinal $\alpha $ for which there exists a well-ordering of $S$ having order type $\alpha $. We will refer to $|S|$ as the cardinality of the set $S$. A cardinal is an ordinal $\kappa $ which has the form $|S|$, for some set $S$.
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