Example 4.7.2.7 (Finite Cardinals). Let $n$ be a nonnegative integer. Then a set $S$ has cardinality $n$ (in the sense of Definition 4.7.2.1) if and only if it has exactly $n$ elements: that is, there exists a bijection $S \simeq \{ 0 < 1 < \cdots < n-1 \} $. In particular, $n$ is a cardinal. We will say that a cardinal $\kappa $ is finite if it arises in this way (that is, if it is the cardinality of a finite set); otherwise, we say that $\kappa $ is infinite.
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