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A set is uncountable if it contains too many items to be counted. "Too many" in this sense means that there are an infinite number of items in the set and that the items in the set cannot be matched up one for one with the natural numbers (i.e. 0, 1, 2, 3, . . . ). For example, the set {1, 2, 4, 8, 16, 32, . . .} is countable because we can match up the natural numbers to this set by noting that the numbers in the latter set are equal to two to the power of the elements in the former set (i.e. 1=2^0, 2=2^1, 4=2^2, 8=2^3, etc.) On the other hand, the real numbers are uncountable, as demonstrated by Cantor's famous diagonalization argument.

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