Cantor's diagonal argument

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Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. Such sets are now known as uncountable sets, and the size of infinite sets is now treated by the theory of cardinal numbers which Cantor began.

The diagonal argument was not Cantor's first proof of the uncountability of the real numbers; it was actually published much later than his first proof, which appeared in 1874. However, it demonstrates a powerful and general technique that has since been used in a wide range of proofs, also known as diagonal arguments by analogy with the argument used in this proof. The most famous examples are perhaps Russell's paradox, the first of Gödel's incompleteness theorems, and Turing's answer to the Entscheidungsproblem.


An uncountable set

Cantor's original proof considers an infinite sequence of the form (s1, s2, s3, ...) where each element si is an infinite sequence of 0s and 1s. Consider any countable listing (for every natural number we associate one and only one element of the set) of these sequences. We might have for instance:

For each m and n let sn,m be the mth element of the nth sequence on the list; so for each n,

It is possible to build a sequence of elements s0 in such a way that its first element is different from the first element of the first sequence in the list, its second element is different from the second element of the second sequence in the list, and, in general, its nth element is different from the nth element of the nth sequence in the list. That is to say, if sm,m is 1, then s0,m is 0, otherwise s0,m is 1. For instance:

(The elements s1,1, s2,2, s3,3, and so on, are here highlighted, showing the origin of the name "diagonal argument". Note that the highlighted elements in s0 are in every case different from the highlighted elements in the table above it.)

Therefore it may be seen that this new sequence s0 is distinct from all the sequences in the list. This follows from the fact that if it were identical to, say, the 10th sequence in the list, then we would have s0,10 = s10,10. In general, if it appeared as the nth sequence on the list, we would have s0,n = sn,n, which, due to the construction of s0, is impossible.

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