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View Diary: Godel, Escher, Bach series:  Introduction and three part invention (170 comments)

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  •  Self Reference (3+ / 0-)
    Recommended by:
    plf515, MD, FrugalGranny

    GEB is a wonderful book, and I still remember it as the book that got me excited about logic.

    However, there is one serious misconception that many readers seem to get from it (even though Hofstadter makes it clear that this is not true) - the misconception is that Goedel's incompleteness theorem has something to do with self-reference.  There are numerous paradoxes that are essentially the same as the key statement that Goedel uses to prove his theorem, dating all the way back to ancient Greece.  The beauty of Goedel is that he showed incompleteness without resorting to self-reference.

    The math behind this isn't that exciting, but he basically did this by mapping all statements (including self-referential ones) to numbers ("Goedel numbering"), so that any self-referential statements are really just properties of numbers.  Thus, any system powerful enough to describe numbers (addition, multiplication), is by its very nature incomplete.  This includes computers and humans.

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