Cantor's diagonal argument

This construction uses a method devised by Cantor that was published in 1878. He used it to construct a bijection between the closed interval and the irrationals in the open interval (0, 1). He first removed a countably infinite subset from each of these sets so that there is a bijection between the remaining uncountable sets. Since there is a bijection between the countably infinite subsets that have been removed, combining the two bijections produces a bijection between the original sets. In set theory, 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 mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers.:20–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, which appeared in 1874.However, it demonstrates a general technique that has since been used in a wide range of proofs, including the first of Gödel's incompleteness theorems and Turing's answer to the Entscheidungsproblem. Diagonalization arguments are often also the source of contradictions like Russell's paradox and Richard's paradox.:27 In his 1891 article, Cantor considered the set T of all infinite sequences of binary digits (i.e. each digit is zero or one).He begins with a constructive proof of the following theorem: The proof starts with an enumeration of elements from T, for example: Next, a sequence s is constructed by choosing the 1st digit as complementary to the 1st digit of s1 (swapping 0s for 1s and vice versa), the 2nd digit as complementary to the 2nd digit of s2, the 3rd digit as complementary to the 3rd digit of s3, and generally for every n, the nth digit as complementary to the nth digit of sn. For the example above, this yields: By construction, s differs from each sn, since their nth digits differ (highlighted in the example).Hence, s cannot occur in the enumeration. Based on this theorem, Cantor then uses a proof by contradiction to show that: The proof starts by assuming that T is countable.Then all its elements can be written as an enumeration s1, s2, ... , sn, ... .Applying the previous theorem to this enumeration produces a sequence s not belonging to the enumeration. However, this contradicts s being an element of T and therefore belonging to the enumeration. This contradiction implies that the original assumption is false. Therefore, T is uncountable.