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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.
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.
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 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.
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.
[[Category:Set theory]]
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]]