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In logic, Richard's paradox is a semantical antinomy of set theory and natural language described first by the French mathematician Jules Richard during 1905. The paradox is ordinarily used to motivate the importance of distinguishing carefully between mathematics and metamathematics. The paradox was also a motivation of the development of predicative mathematics. == Description == The original statement of the paradox, due to Richard (1905), has a relation to Cantor's diagonal argument on the uncountability of the set of real numbers. 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, while the phrase "the capital of England" does not define a real number. Thus there is an infinite list of English phrases (such that each phrase is of finite length, but lengths vary in the list) that define real numbers unambiguously; arrange this list by length and then order lexicographically (in dictionary order), so that the ordering is canonical. This yields an infinite list of the corresponding real numbers: ''r''1, ''r''2, ... . 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''''n'' is not 1, and the ''n''th decimal place of ''r'' is 2 if the ''n''th decimal place of ''r''''n'' is 1. The preceding two paragraphs are an expression in English that unambiguously defines a real number ''r''. Thus ''r'' must be one of the numbers ''r''''n''. However, ''r'' was constructed so that it cannot equal any of the ''r''''n''. This is the paradoxical contradiction. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Richard's paradox」の詳細全文を読む スポンサード リンク
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