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We've seen something similar before: we know that the collection of all languages is uncountable (like the reals), while the collection of Turing-recognizable languages is countable (like the integers). Our conclusion regarding Kolmogorov complexity is that, even though the set of true statements is countable, and the set of provable statements is countable, for any axiomatic system A, the set of true but not provable statements is a constant fraction of all the statements. |