We construct models of the integers, to yield: witnessing, independence and separation results for weak systems of bounded induction.
We show that the arithmetical theory T 0 2 + ˆ Σ b 1-IN D |x| 5 , formalized in the language of Buss, i.e. with x/2 but without the M SP function x/2 y , does not prove that every nontrivial divisor of a power of 2 is even. It follows that this theory proves neither N P = coN P nor S 0 2. Some arithmetical theories are not merely weak but very weak, in the… (More)