On the limit existence principles in elementary arithmetic and -consequences of theories

We study the arithmetical schema asserting that every eventually decreasing elementary recursive function has a limit. Some other related principles are also formulated. We establish their relationship with restricted parameter-free induction schemata. We also prove that the same principle, formulated as an inference rule, provides an axiomatization of the Σ 2 -consequences of I Σ 1 . these results we show that ILM is the logic of Π 1 -conservativity of any reasonable extension of parameter-free Π 1 -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 Π 1 -conservativity of primitive recursive arithmetic properly extends ILM. third part of the paper we give an ordinal classification of Σ n 0 -consequences 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 Π -classes of sentences (usually Π 1 1 or Π 2 0 ).