Table of contents |
2 The theorem 3 The proof 4 Real algebraic numbers and real transcendental numbers |
Contrary to what most mathematicians believe, Georg Cantor's first proof that the set of all real numbers is uncountable was not his famous diagonal argument, and did not mention decimal expansions or any other numeral system. The theorem and proof below were found by Cantor in December 1873, and published in 1874 in Crelle's Journal, more formally known as Journal für die Reine und Angewandte Mathematik (German for Journal for Pure and Applied Mathematics). Cantor discovered the diagonal argument in 1877.
Suppose a set R is
The proof begins by assuming some sequence x1, x2, x3, ... has all of R as its range. Define two other sequences as follows:
In the same paper, published in 1874, Cantor showed that the set of all real algebraic numbers is countable, and inferred the existence of transcendental numbers as a corollary. That corollary had earlier been proved by quite different methods by Joseph Liouville.The misconception
The theorem
Then R is not countable.
The proof
The two monotone sequences a and b move toward each other. By the "gaplessness" of R, some point c must lie between them. The claim is that c cannot be in the range of the sequence x, and that is the contradiction. If c were in the range, then we would have c = xi for some index i. But then, when that index was reached in the process of defining a and b, then c would have been added as the next member of one or the other of those two sequences, contrary to the assumption that it lies between their ranges.Real algebraic numbers and real transcendental numbers