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Cantor's infinity diagonalisation proof. Diagram showing how the German mathematician Georg Cantor (1845-1918) used a diagonalisation argument in 1891 to show that there are sets of numbers that are uncountable. The set of natural numbers (1, 2, 3..) is infinite, but can be shown to be countable (see A900/141). The set of real numbers (infinite decimals), however, is not countable, and is considered to be of a higher order of infinity. This is shown by constructing an array of infinite decimals to form the diagonal number (green). By subtracting one from each digit of the green number, a new number (red) has been formed that can never be found in the original infinite array. The relative sizes of such infinite sets are described by cardinal numbers.
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