Cantor’s theorem

Cantor’s theorem, in set theory, the theorem that the cardinality (numerical size) of a set is strictly less than the cardinality of its power set, or collection of subsets. In symbols, a finite set S with n elements contains 2ⁿ subsets, so that the cardinality of the set S is n and its power set P(S) is 2ⁿ. While this is clear for finite sets, no one had seriously considered the case for infinite sets before the German mathematician Georg Cantor—who is universally recognized as the founder of modern set theory—began working in this area toward the end of the 19th century.

The 1891 proof of Cantor’s theorem for infinite sets rested on a version of his so-called diagonalization argument, which he had earlier used to prove that the cardinality of the rational numbers is the same as the cardinality of the integers by putting them into a one-to-one correspondence. The notion that, in the case of infinite sets, the size of a set could be the same as one of its proper subsets was not too surprising, as before Cantor almost everyone assumed that there was only one size for infinity. However, Cantor’s proof that some infinite sets are larger than others—for example, the real numbers are larger than the integers—was surprising, and it initially met with great resistance from some mathematicians, particularly the German Leopold Kronecker. Furthermore, Cantor’s proof that the power set of any set, including any infinite set, is always larger than the original set led him to create an ever increasing hierarchy of cardinal numbers, ℵ₀, ℵ₁, ℵ₂…, known as transfinite numbers. Cantor proposed that there is no transfinite number between the first transfinite number ℵ₀, or the cardinality of the integers, and the continuum (c), or the cardinality of the real numbers; in other words, c = ℵ₁. This is now known as the continuum hypothesis, and it has been shown to be an undecidable proposition in standard set theory.