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)
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