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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 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 [[order]]ed it is immediate that any countable set can also be ordered. Uncountable sets can only be ordered through use of the [[axiom of choice]].

Since the natural numbers are [[order]]ed it is immediate that any countable set can also be ordered. Arbitrary uncountable sets can only be ordered through use of the [[axiom of choice]], however certain uncountable sets (such as the [[real number]]s) come with their own ordering.

[[Category:Set theory]]

[[Category:Set theory]]