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  Goedel Incompleteness Theorem leaves open a way around it, 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 multiple solutions, e.g. non-recursive tilings. 
Leonid A. Levin