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Different Kinds of Infinity

Different Kinds of Infinity


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Types of infinity

Are all infinite sets of numbers the same size? No. The set of irrationals and the set of reals are not countable. There is no way that you can lay them out so there is a one-to-one correspondence with the natural numbers. This means that there are different types of infinity. The countable sets of natural numbers and rationals are smaller than the sets of irrationals and reals.

In elementary set theory, Cantor's theorem states that, for any set A, the set of all subsets of A (the power set of A) has a strictly greater cardinality than A itself. For finite sets, Cantor's theorem can be seen to be true by a much simpler proof than that given below, since in addition to subsets of A with just one member, there are others as well, and since n < 2n for all natural numbers n. But the theorem is true of infinite sets as well. In particular, the power set of a countably infinite set is uncountably infinite. The theorem is named for German mathematician Georg Cantor, who first stated and proved it.

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