## The interpretability logic of Peano arithmetic (1990)

Venue: | The Journal of Symbolic Logic |

Citations: | 16 - 0 self |

### BibTeX

@ARTICLE{Berarducci90theinterpretability,

author = {Alessandro Berarducci and Alessandro Berarducci and To Silvia},

title = {The interpretability logic of Peano arithmetic},

journal = {The Journal of Symbolic Logic},

year = {1990},

volume = {55},

pages = {1059--1089}

}

### Years of Citing Articles

### OpenURL

### Abstract

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### Citations

53 |
Provability Interpretations of Modal Logic
- Solovay
- 1976
(Show Context)
Citation Context ...his paper we use the language of modal logic to give an axiomatic treatment of interpretability similar to and extending the corresponding axiomatic treatment of provability carried out by Solovay in =-=[16]-=-. We need to fix a 'base' theory T satisfying the hypothesis of Orey's theorem and such that T does not prove false cY-assertions (e.g. PA or ZF).Any such theory will work, but for simplicity we take ... |

28 | consistency statements, and interpretations - Cuts - 1985 |

20 |
Provability logics for relative interpretability
- Jongh, Veltman
(Show Context)
Citation Context ... with respect to interpretations with base theory PA, gives a positive solution to a conjecture of Visser and makes essential use of the semantical analysis of ILM carried out by De Jongh and Veltman =-=[5]-=-, and simplified by Visser [18]. (We need Visser's simplified semantics in our proof.) This paper is an extended version of a preliminary manuscript [2] containing a proof of the completeness theorem.... |

10 |
The logic of relative interpretability over Peano arithmetic (in Russian
- Shavrukov
- 1988
(Show Context)
Citation Context ... (We need Visser's simplified semantics in our proof.) This paper is an extended version of a preliminary manuscript [2] containing a proof of the completeness theorem. A recent preprint of Shavrukov =-=[13]-=- also contains a proof of the completeness theorem. The two proofs are quite different and were obtained independently from each other. This paper is organized as follows. $2 contains some preliminary... |

5 |
Nonarithmeticity of truth predicate logics of provability, Dokl. Akad. Nauk SSSR 284
- Artemov
- 1985
(Show Context)
Citation Context ...ntroduction of quantifiers over numbers (as in OVxA(x) -+ VxOA(x)). However Vardanyan 1171 proved that this leads to an undecidable set of PA-valid formulas (even without the D-operator), and Artemov =-=[I]-=- showed that the corresponding set of w-valid formulas is not even arithmetical. In [3] we proved that Vardanyan's undecidability result still holds even if we restrict the interpretation of the atomi... |

5 |
Translatability and relative consistency
- Friedman
- 1979
(Show Context)
Citation Context ...Orey's theorem T would prove the consistency of itself, contradicting Godel's second incompleteness theorem.) For finitely axiomatizable sequential theories there is a theorem of Harvey Friedman (see =-=[7]-=- or [14]) that gives a similar characterization of interpretability in terms of consistency: let T and S be finitely axiomatized sequential theories, then T interprets S iff the theory Id, + EXP prove... |

3 |
Nonstandard, models. and related developments, in the ,work of Harvey Friedman
- Smorynski
- 1985
(Show Context)
Citation Context ...theorem T would prove the consistency of itself, contradicting Godel's second incompleteness theorem.) For finitely axiomatizable sequential theories there is a theorem of Harvey Friedman (see [7] or =-=[14]-=-) that gives a similar characterization of interpretability in terms of consistency: let T and S be finitely axiomatized sequential theories, then T interprets S iff the theory Id, + EXP proves that t... |

2 |
Some applications of cut elimination. In: Handbook of Mathematical Logic
- SCHWICHTENBERG
- 1977
(Show Context)
Citation Context ...ction). The following is provable in PA: for every k and n, PA proves (for every C:-sentence 4, if PA, I- 4,then 4 is true'. PROOF. The usual effective proof of Gentzen's cut elimination theorem (see =-=[12]-=-) is formalizable in PA. A consequence of this theorem is that if a formula x has a proof in first order logical calculus (from no axioms), then it has a proof such that all the formulas appearing in ... |

1 |
The interpretability logic of Peano arithmetic (preliminary version
- BERARDUCCI
- 1988
(Show Context)
Citation Context ...ysis of ILM carried out by De Jongh and Veltman [5], and simplified by Visser [18]. (We need Visser's simplified semantics in our proof.) This paper is an extended version of a preliminary manuscript =-=[2]-=- containing a proof of the completeness theorem. A recent preprint of Shavrukov [13] also contains a proof of the completeness theorem. The two proofs are quite different and were obtained independent... |

1 |
The unprovability of consistency
- GEORGEBOOLOS
- 1979
(Show Context)
Citation Context ...models (hence L is decidable): THEOREM4.3 (Modal completeness of L). If L tf A, then there is a finite L-model V, with root b say, such that b 111A. For a proof of this theorem we refer the reader to =-=[4]-=- or [16]. 4.2. Simplified ILM-models. DEFINITION 4.4. A simplijied ILM-frame is an L-frame (V, R, b) together with an additional relation S with the following properties: 1. S is a transitive, reflexi... |

1 | Arithmetization of metamathematics in a general setting - SOLOMONFEFERMAN - 1960 |

1 | Some results on interpretability, Proceedings from 5th Scandinavian logic symposium - LINDSTROM - 1979 |

1 |
Semantical closure and non-finite axiomatizability. I, Infinitistic methods (proceedings of the symposium on foundations of mathematics
- MONTAGUE
- 1959
(Show Context)
Citation Context ...of) an axiom of T. An application of the partial truth definitions (and the cut elimination theorem) is that PA proves the C:-soundness (hence the consistency) of every finite subtheory of itself (cf =-=[lo]-=-). THEOREM2.6 (Reflection). The following is provable in PA: for every k and n, PA proves (for every C:-sentence 4, if PA, I- 4,then 4 is true'. PROOF. The usual effective proof of Gentzen's cut elimi... |