  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.
