Cantor's diagonal argument
proof technique in set theory
Cantor's diagonal argument (among various similar names) is a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers – informally, that there are sets which in some sense contain more elements than there are positive integers. Such sets are now called uncountable sets, and the size of infinite sets is treated by the theory of cardinal numbers, which Cantor began.
Georg Cantor published this proof in 1891, but it was not his 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.
Uncountable set
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 lemma:
If s1, s2, ... , sn, ... is any enumeration of elements from T, then an element s of T can be constructed that doesn't correspond to any sn in the enumeration.
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