 This phrase defines a real number, so it must be among the enumerated phrases, say number \(k\) in this enumeration. But, at the same time, by definition, it differs from the number denoted by the \(k\)th phrase in the \(k\)th decimal place. Thus we have a contradiction. The defining phrase is obviously impredicative. The particular construction employed in this paradox is called diagonalisation. Diagonalisation is a general construction and proof method originally invented by Georg Cantor (1891) to prove the uncountability of the power set of the natural numbers. It was also used as a basis for Cantor’s paradox, one of the set-theoretic paradoxes to be considered next.