Foundations of Mathematics

Set Theory

What does it mean if a set is countable?

If a set is countable (or denumerable), it means that it is finite. This also means that the set’s members can be matched in a one-to-one correspondence, in which each element in one set is matched exactly with one element in the second, and vice versa, with all the natural numbers, or with a subset of the natural numbers. Mathematicians often say, “A and B are in one-to-one correspondence,” or “A and B are bijective.” (For more about one-to-one correspondence, see “Math Basics.”) In set theory, all finite sets are considered to be countable, as are all subsets of the natural numbers and integers. But sets such as real numbers, points on a line, and complex numbers are not countable.


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