Richard's paradox

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A set of numbers, S, may be defined as follows. Write in alphabetical order all permutations of pairs of letters of the alphabet, followed by all triples, and so on (counting a space and punctuation marks as letters). Cross out all combinations that do not refer to a number. S is the sequence whose members are U1, the first number referred to by such a permutation, U2, the second, and so on. That is, S contains all the numbers definable by finitely many words. Then consider the following sentence: ‘Let p be the digit in the nth place of the nth number in S, and form a number having p+1 in its nth place if p is not 8 or 9, and 0 otherwise.’ Then this phrase refers to a number, N, that cannot be in the sequence S, since it differs from any member of this sequence somewhere. It differs from the nth member of S in the nth place. Yet the sentence in quotation marks is one of the permutations that occurs in the original list defining the members of S: N was defined by finitely many words, and so must be in the sequence S. The paradox is a member of the Liar family of semantic paradoxes. The argument generating the paradox is a diagonal argument.

Subjects: Philosophy.

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