Otherwise, if all the elements in a set can be represented by a at most n symbols (finite Kolmogorov complexity), I could count them by creating a n dimensional pairing function. Or atleast, that is my assumption.


Any thoughts?


Edit: as @Carl has pointed out the correct term for sequences that have no finite representation is Kolmogorov Random not infinite as I used in my title.



Do all uncountable sets contain elements with infinite Kolmogorov complexity?
Contrary to what some of the commenters said, there's no great difficulty in generalizing Kolmogorov complexity to infinite strings. For example, given a language L⊆{0,1}\*, we could let K(L) be the length of the shortest program that decides L, or K(L)=∞ if L is undecidable. (Or we could also talk about programs that *recognize* L, in which case even the language HALT would have a finite complexity.)


In either case, the OP's intuition is essentially correct: if a set S of languages is uncountable, then since there are only countably many decidable or recognizable languages, clearly there must exist a language L∈S such that K(L)=∞.


