Abstract
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 ∑2-consequences of I ∑1. Using these results we show that ILM is the logic of Π1 -conservativity of any reasonable extension of parameter-free Π1-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 Π1-conservativity of primitive recursive arithmetic properly extends ILM. In the third part of the paper we give an ordinal classification of ∑n0-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 Π11 or Π20).
| Original language | English |
|---|---|
| Pages (from-to) | 56-74 |
| Number of pages | 19 |
| Journal | Annals of Pure and Applied Logic |
| Volume | 136 |
| Issue number | 1-2 |
| DOIs | |
| Publication status | Published - Oct 2005 |
Bibliographical note
Funding Information:We would like to thank J. Joosten for useful comments and proof reading and G. Mints for a kind permission to include one of his results. Special thanks go to an anonymous referee whose remarks eventually led us to a proof of Theorem 7.Infact, the referee proved that a version of the limit rule with arbitrary Σ2 side formulas provides an axiomatization of Σ2-consequences of I Σ1, and we were later able to improve this result to the original rule without side formulas by dealing with the notion of special Σ2-formula. The first author was supported in part by the Russian Foundation for Basic Research.
Funding
We would like to thank J. Joosten for useful comments and proof reading and G. Mints for a kind permission to include one of his results. Special thanks go to an anonymous referee whose remarks eventually led us to a proof of Theorem 7.Infact, the referee proved that a version of the limit rule with arbitrary Σ2 side formulas provides an axiomatization of Σ2-consequences of I Σ1, and we were later able to improve this result to the original rule without side formulas by dealing with the notion of special Σ2-formula. The first author was supported in part by the Russian Foundation for Basic Research.
Keywords
- Conservativity
- Elementary arithmetic
- Inference rule
- Interpretability logic
- Ordinal analysis
- Parameter-free induction
- Reflection principles