The discussion revolves around the Kolmogorov complexity of irrational numbers, particularly focusing on whether noncomputable irrational numbers possess infinite complexity. Participants explore methods to determine the complexity and examine examples that illustrate the nuances of noncomputable numbers.
Participants express differing views on the implications of the example presented, indicating that the discussion remains unresolved regarding the nature of randomness among noncomputable numbers and their Kolmogorov complexity.
The discussion includes assumptions about the definitions of Kolmogorov complexity and noncomputable numbers, which may not be universally agreed upon. The implications of the example regarding information requirements are also not fully explored.
cragar said:How would I figure out the Kolmogorov complexity of an irrational number?
Could I just look at the shortest formula to compute that number.
If it is an uncomputable real, does it have infinite complexity?
SteveL27 said:I saw an interesting example online the other day. Take a noncomputable number and interperse 0's in every other digit position. Then for each finite-length initial segment, it takes just a little more than half as much information to compute the number with the 0's in every other position as it does to compute the original number. So not all noncomputable numbers are equally random. It's tricker than I'd realized.