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Math Help - Russell's Paradox

  1. #1
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    Russell's Paradox

    Question: Can there exist a book that refers to all those books and only those books that do not refer to themselves?

    I have reviewed the example in the book, and according to Russell's Paradox, I think the answer is neither yes nor no. If the book refers to itself, then it does not exist. However, if the book does not refer to itself, then it exists.

    Is my thought process correct?
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  2. #2
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    Re: Russell's Paradox

    The answer is no.

    There does not exist a book that refers to all and only those books that do not refer to themselves.

    Suppose there is a B that refers to all an only those X that do not refer to X.

    Then B refers to B if and only if B does not refer to B, which is a contradiction.

    So there is no B that refers to all and only those X that do not refer to X.

    It doesn't matter what 2-place property you choose ('refer', 'knows', 'shaves' 'is a member of', whatever). For any 2-place property P, we have

    There does not exist a b such that for all x, Pbx if and only if ~Pxx.
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  3. #3
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    Re: Russell's Paradox

    Thank you for your help.
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