The author is referring to something called Chaitin incompleteness. https://en.wikipedia.org/wiki/Kolmogorov_complexity#Chaitin's_incompleteness_theorem https:

The author is referring to something called Chaitin incompleteness. https://en.wikipedia.org/wiki/Kolmogorov_complexity#Chaitin's_incompleteness_theorem https://en.wikipedia.org/wiki/Kolmogorov_complexity#Chaitin'... Of course trivially some KC can be proven, ex a language with 1 or 0 characters that is interpreted to a specific string. Or to prove KC(x) where the compressed value has length N and you can list out all the results for all strings of length less than N, and they don't equal x, proves KC(x)=N. The interesting number paradox (Berry's paradox) is more related to Chaitin incompleteness. Basically, given a language there’s some code which enumerates proofs that KC of a string is more than some constant L, and returns the first one it finds. If the constant L is large enough, it becomes larger than the entire proof generating code. So the proof generating code will never find a proof of any KC larger than L. It's interesting to think about that the language gets more complex, proofs for larger strings become possible. And what it would mean for the languages to keep getting more complex indefinitely. it's a similar train of thought to busy beaver numbers and how systems of logic (PA,ZFC) become independent to values like BB(745), and what it could mean to have more and more advanced types of logic which don't become independent until some high target n.