Cantor's diagonal argument - Real numbers

Cantor's diagonal argument is a proof devised by Georg Cantor to demonstrate that the real numbers are not countably infinite. (It is also called the diagonalization argument or the diagonal slash argument or the diagonal method.) The diagonal argument was not Cantor's first proof of the uncountability of the real numbers, but was published three years after his first proof. His original argument did not mention decimal expansions, nor any other numeral system. Since this technique was first used, si ...

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• Cantor's diagonal argument - Real numbers
• Cantor's diagonal argument - Why this does not work on integers
• Cantor's diagonal argument - General sets

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Cantor's original proof shows that the interval [0,1] is not countably infinite. The proof by contradiction proceeds as follows: Assume (for the sake of argument) that the interval [0,1] is countably infinite. We may then enumerate all numbers in this interval as a sequence, ( r1, r2, r3, ... ) We already know that each of these numbers may be represented as a decimal expansion. We arrange the numbers in a list (they do not need to be in orde ...

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Cantor's diagonal argument, Cantor's diagonal argument - Real numbers, Cantor's diagonal argument - Why this does not work on integers, Cantor's diagonal argument - General sets

Read more here: » Cantor's diagonal argument: Encyclopedia II - Cantor's diagonal argument - Real numbers