Editing Richard's paradox (section)

== Description ==

The original statement of the paradox, due to Richard (1905), is strongly related to [[Cantor's diagonal argument]] on the uncountability of the set of [[real number]]s.

The paradox begins with the observation that certain expressions of natural language define real numbers unambiguously, while other expressions of natural language do not. For example, "The real number the integer part of which is 17 and the ''n''th decimal place of which is 0 if ''n'' is even and 1 if ''n'' is odd" defines the real number 17.1010101... = 1693/99, whereas the phrase "the capital of England" does not define a real number, nor the phrase "the smallest positive integer not definable in under sixty letters" (see [[Berry's paradox]]).

There is an infinite list of English phrases (such that each phrase is of finite length, but the list itself is of infinite length) that define real numbers unambiguously. We first arrange this list of phrases by increasing length, then order all phrases of equal length [[Lexicographical order|lexicographically]], so that the ordering is [[Canonical form|canonical]]. This yields an infinite list of the corresponding real numbers: ''r''<sub>1</sub>, ''r''<sub>2</sub>, ... .  Now define a new real number ''r'' as follows. The integer part of ''r'' is 0, the ''n''th decimal place of ''r'' is 1 if the ''n''th decimal place of ''r''<sub>''n''</sub> is not 1, and the ''n''th decimal place of ''r'' is 2 if the ''n''th decimal place of ''r''<sub>''n''</sub> is 1.

The preceding paragraph is an expression in English that unambiguously defines a real number ''r''. Thus ''r'' must be one of the numbers ''r''<sub>''n''</sub>. However, ''r'' was constructed so that it cannot equal any of the ''r''<sub>''n''</sub> (thus, ''r'' is an [[undefinable number]]). This is the paradoxical contradiction.