Proof Idea: The proof of this result is modeled on a self-referential construction used in Berry's paradox. We firstly obtain a program which enumerates the proofs within S and we specify a procedure P which takes as an input an integer L and prints the strings x which are within proofs within S of the statement K(x) ≥ L. By then setting L to greater than the length of this procedure P, we have that the required length of a program to print x as stated in K(x) ≥ L as being at least L is then less than the amount L since the string x was printed by the procedure P. This is a contradiction. So it is not possible for the proof system S to prove K(x) ≥ L for L arbitrarily large, in particular, for L larger than the length of the procedure P, (which is finite).
Proof Idea: The proof of this result is modeled on a self-referential construction used in Berry's paradox. We firstly obtain a program whi?ch enumerates the proofs within S and we specify a procedure P which takes as an input an integer L and prints the strings x which are within proofs within S of the statement K(x) ≥ L. By then setting L to greater than the length of this procedure P, we have that the required length of a program to print x as stated in K(x) ≥ L a?s being at least L is then less than the amount L since the string x was printed by the procedure P. This is a contradiction. So it is not possible for the proof system S to pr,ove K(x) ≥ L for L arbitrarily large, in particular, for L larger .than the length of the procedure P, (which is finite).