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A [[set]] is said to be '''finite''' if its elements can be labeled with the numbers from \(1\) to \(n\), for some natural number \(n\). A set that isn't finite is said to be [[infinite]].
More precisely, a set \(S\) is finite when there exists \(n\in \mathbb N\) and a [https://en.wikipedia.org/wiki/Bijection bijective function] (a one-to-one correspondence) \(f:S\to\{<n\}\), where \(\{<n\}\) denotes the set of naturals less than \(n\). The unique natural for which this holds is called its [[cardinality]], although this concept may be defined in greater generality. Perhaps the simplest finite set is the [[empty set]] \(\varnothing\), whose cardinality is [[0]]. Any singleton set \(\{a\}\) is finite and has cardinality [[1]].
An [[ordinal]] is called '''finite''' when it's the [[order type]] of a finite [[well-ordered set]]. It can be proven that these ordinals correspond precisely to the order types of the sets \(\{<n\}\). This allows for the identification of finite ordinals with natural numbers.
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