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. 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.
Read more about Cantor's Diagonal Argument: An Uncountable Set, General Sets
Famous quotes containing the word argument:
“Argument is conclusive ... but ... it does not remove doubt, so that the mind may rest in the sure knowledge of the truth, unless it finds it by the method of experiment.... For if any man who never saw fire proved by satisfactory arguments that fire burns ... his hearer’s mind would never be satisfied, nor would he avoid the fire until he put his hand in it ... that he might learn by experiment what argument taught.”
—Roger Bacon (c. 1214–1294)