Unbounded arithmetic

When comparing the different axiomatizations of bounded arithmetic and Peano arithmetic, it becomes clear that there are similarities between the fragments of these theories. In particular, it is tempting to draw an analogy between the hierarchies of bounded arithmetic and Peano arithmetic. However, one cannot deny that there are essential and deeply rooted differences and the most one can claim is a weak analogy between these hierarchies. The following quote by Kaye expresses this argument in an elegant way: "Many authors have emphasized the analogies between the fragments Sigma(b)(n)-IND of IDelta0+(Vx)(xlog x) exists) and the fragments ISigma(n) of Peano arithmetic. Sometimes this is helpful, but often one feels that the bounded hierarchy of theories is of a rather different nature and new techniques must be developed to answer the key questions concerning them." In this paper, we propose a (conjectured) hierarchy for Peano arithmetic which is much closer to that of bounded arithmetic than the existing one. In light of this close relation, techniques developed to establish properties of the new hierarchy should carry over naturally to the (conjectured) hierarchy of bounded arithmetic. As the famous P vs. NP problem is related to the collapse of the hierarchy of bounded arithmetic, the new hierarchy may prove particularly useful in solving this famous problem