Well-behaved principles alternative to bounded induction

We introduce some Π1–expressible combinatorial principles which may be treated as axioms for some bounded arithmetic theories. The principles, denoted Sk(Σn, length log k) and Sk(Σn, depth log k) (where ‘Sk’ stands for ‘Skolem’), are related to the consistency of Σn–induction: for instance, they provide models for Σn–induction. However, the consistency is expressed indirectly, via the existence of evaluations for sequences of terms. The evaluations do not have to satisfy Σn–induction, but must determine the truth value of Σn statements. Our principles have the property that Sk(Σn, depth log k) proves Sk(Σn+1, length log k). Additionally, Sk(Σn, length log k−2) proves Sk(Σn+1, length log k). Thus, some provability is involved where conservativity is known in the case of Σn induction on an initial segment and induction for higher Σm classes on smaller segments. ∗Research supported in part by The State Committee for Scientific Research (Poland), KBN, grant number 5 PO3A 037 20.