On Effective Speed-up and Long Proofs of Trivial Theorems in Formal Theories

Abstract

In this note we give a very simple proof which shows that in many interesting formal mathematical theories, axiomatizable as well as decidable ones, for every given formalization we can effectively find infinite subsets of trivially true theorems which require as long proofs in the given formalism as the hardest theorems of the theory. Thus showing that for these theories every formalism is doomed to be blind to the triviality of infinite sets of theorems, which can be found effectively. Furthermore, it follows that for all (sufficiently large) constructable tape and time bounds there exists sets whose recognition can be effectively speeded up on infinite subsets and that such sets appear naturally, thus showing that for many concrete problems, every algorithm can be effectively sped-up on infinite subsets.