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A set X is countable if and only if there is a bijection from X to the set of [[Natural Numbers]].
A [[set]] X is '''countable''' if and only if there is a [[bijection]] from X to a subset set of [[Natural Numbers|natural numbers]]. Countable sets include finite sets, the set of [[integer]]s, and the set of [[rational number]]s.
[[Category: Mathematics]]
There is no smallest infinite countable set. Indeed, the set of natural numbers is in bijection with the natural numbers without 0: subtracting 1 from every number gives a bijection from the first set to the second. Repeating this process shows that for any initial segment of the natural numbers (such as {1, 2, ..., n}), we have a bijection between the set of natural numbers and the set of natural numbers without this segment. The bijection is simply subraction by n.
Since the natural numbers are [[Well-Ordering Theorem|well-ordered]] it is immediate that any countable set can also be well-ordered. Arbitrary uncountable sets can only be well-ordered through use of the [[axiom of choice]].
[[Category:Set Theory]]