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Mon, 08/31/2009 - 17:16 — SomNambul
Authors: Leonid A. Levin
Subjects: Computational Complexity
link: http://arxiv.org/abs/cs/0203029
Abstract

There appears to be a loophole in Goedel Incompleteness Theorem, vaguely
perceived for a long time but not clearly identified. (Thus, Goedel believed
informal arguments can answer any math question.) Closing this loophole does
not seem obvious and involves Kolmogorov complexity. (This is unrelated to,
well studied before, complexity quantifications of the usual Goedel effects.)

I consider extensions U of the universal partial recursive predicate (or,
say, Peano Arithmetic). I prove that any U either leaves an n-bit input
(statement) unresolved or contains nearly all information about the n-bit
prefix of any r.e. real r (which is n bits for some r). I argue that creating
significant information about a SPECIFIC math sequence is impossible regardless
of the methods used. Similar problems and answers apply to other unsolvability
results for tasks allowing non-unique solutions, e.g. non-recursive tilings.

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