A special case of Cantor's theorem proves that the set of all real numbers R cannot be enumerated by natural numbers. These sets have in common the cardinal number | N| = ℵ 0 with the usual order 0 | S|. Examples of countably infinite sets are the natural numbers, the even numbers, the prime numbers, and also all the rational numbers, i.e., the fractions. Every infinite set which can be enumerated by natural numbers is the same size (cardinality) as N, and is said to be countable. As this assumption cannot be proved from first principles it has been introduced into axiomatic set theory by the axiom of infinity, which asserts the existence of the set N of natural numbers. Set theory as conceived by Georg Cantor assumes the existence of infinite sets.
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