Statements such as K(n)> m
cannot be proven above a certain value of m,
though they are true for infinitely many integers n.
Suppose that for infinitely many m, there is an integer xm such that K(xm)> m is provable. The value of xm can be determined by an algorithm that enumerates all theorems. This algorithm has only m as input. Therefore: K(xm)<2log(m)+ O(1) (using a doubling code as prefix code for m). We get: m < K(xm)<2log(m)+ O(1), which is absurd as soon as m is large enough.