Algorithmic information applied to mathematicsKolmogorov complexity is not computable! But... Kolmogorov complexity appears as an ideal notion that can be proven to be incomputable.
However,
it is a fundamental mathematical notion,
and it can be approximated for practical (and realistic) use.
Algorithmic probability and algorithmic information Probability can be defined in purely algorithmic terms.
This, however, requires a distinction between two versions of Kolmogorov complexity.
Randomness What is randomness? The only sound definition of randomness is based on Kolmogorov complexity.
Deductive reasoning Complexity arguments can be used to prove theorems.
Gödel’s theorem revisited Gödel’s theorem is one of the most important theorems in mathematics. It sets limits to what any formal theory can prove (and indirectly, to what AI can achieve). Thanks to Kolmogorov complexity, Gödel’s theorem becomes almost trivial, and much easier to grasp.