Posts Tagged 'Cantor'

Cantor’s Theorem and the Unending Hierarchy of Infinities

The power set of the set {x,y,z}, containing all its subsets, has 2^3=8 elements. Image from Wikimedia Commons.

In 1891, Georg Cantor published a seminal paper, U”ber eine elementare Frage der Mannigfaltigkeitslehren — On an elementary question of the theory of manifolds — in which his “diagonal argument” first appeared. He proved a general theorem which showed, in particular, that the set of real numbers is uncountable, that is, it has cardinality greater than that of the natural numbers. But his theorem is much more general, and it implies that the set of cardinals is without limit: there is no greatest order of infinity.

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