Lifting Independence Results in Bounded Arithmetic

We investigate the problem how to lift the non - 8\Sigma b 1 (ff) - conservativity of T 2 2 (ff) over S 2 2 (ff) to the expected non - 8\Sigma b i (ff) - conservativity of T i+1 2 (ff) over S i+1 2 (ff), for i ? 1. We give a non-trivial refinement of the "lifting method" developed in [4, 8], and we prove a sufficient condition on a 8\Sigma b 1 (f)-consequence of T2 (f) to yield the non-conservation result. Further we prove that Ramsey's theorem, a 8\Sigma b 1 (ff) - formula, is not provable in T 1 2 (ff), and that 8\Sigma b j (ff) - conservativity of T i+1 2 (ff) over T i 2 (ff) implies 8\Sigma b j (ff) - conservativity of the whole T2 (ff) over T i 2 (ff), for any