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Provability in finite subtheories of PA and relative interpretability: a modal investigation. (English) Zbl 0623.03023
The author introduces bimodal logics extending the logic of provability GL with a new modality $$\Delta$$ which characterizes the provability predicate for any sufficiently rich finitely axiomatizable subtheory of Peano Arithmetic PA. Kripke semantics for two of these logics are given together with a uniform arithmetical completeness theorem. The De Jongh- Sambin fixed point theorem is extended to these logics. Lastly, bimodal logic of provability is used to study relative interpretability of theories $$PA+\theta$$ ($$\theta$$ is an arithmetical sentence).
Reviewer: S.Artemov

##### MSC:
 03B45 Modal logic (including the logic of norms) 03F07 Structure of proofs 03F30 First-order arithmetic and fragments 03F25 Relative consistency and interpretations
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