On the limit existence principles in elementary arithmetic and related topics

Abstract

We study the arithmetical schema asserting that every eventually decreasing primitive recursive function has a limit. Some other related principles are also formulated. We establish their relationship with restricted parameter-free induction schemata. Using these results we show that ILM is the logic of II1-conservativity of any reasonable extension of parameter-free II1-induction schema. This result, however, cannot be much improved: by adapting a theorem of D. Zambella and G. Mints we show that the logic of II1-conservativity of primitive recursive arithmetic properly extends ILM. In the third part of the paper we give an ordinal classification of Σn-inconsequences of the standard fragments of Peano arithmetic in terms of reflection principles. This is interesting in view of the general program of ordinal analysis of theories, which in the most standard cases classifies II-classes of sentences (usually II1/1 or II0/2).