Countable

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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. Countable sets include finite sets, the set of integers, and the set of rational numbers.

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.

Since the natural numbers are 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.

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