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Shavrukov's Theorem on the Subalgebras of Diagonalizable Algebras for Theories Containing IΔ0+exp

Domenico Zambella

Notre Dame Journal of Formal Logic 35 (1):147-157 (1994)

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Interpretability over peano arithmetic.Claes Strannegård - 1999 - Journal of Symbolic Logic 64 (4):1407-1425.details We investigate the modal logic of interpretability over Peano arithmetic. Our main result is a compactness theorem that extends the arithmetical completeness theorem for the interpretability logic ILM ω . This extension concerns recursively enumerable sets of formulas of interpretability logic (rather than single formulas). As corollaries we obtain a uniform arithmetical completeness theorem for the interpretability logic ILM and a partial answer to a question of Orey from 1961. After some simplifications, we also obtain Shavrukov's embedding theorem for Magari (...) Download Export citation Bookmark
Provability logic-a short introduction.Per Lindström - 1996 - Theoria 62 (1-2):19-61.details Download Export citation Bookmark 9 citations
Franco Montagna’s Work on Provability Logic and Many-valued Logic.Lev Beklemishev & Tommaso Flaminio - 2016 - Studia Logica 104 (1):1-46.details Franco Montagna, a prominent logician and one of the leaders of the Italian school on Mathematical Logic, passed away on February 18, 2015. We survey some of his results and ideas in the two disciplines he greatly contributed along his career: provability logic and many-valued logic. Download Export citation Bookmark
(1 other version)Substitutions of Σ10-sentences: explorations between intuitionistic propositional logic and intuitionistic arithmetic.Albert Visser - 2002 - Annals of Pure and Applied Logic 114 (1-3):227-271.details This paper is concerned with notions of consequence. On the one hand, we study admissible consequence, specifically for substitutions of Σ 1 0 -sentences over Heyting arithmetic . On the other hand, we study preservativity relations. The notion of preservativity of sentences over a given theory is a dual of the notion of conservativity of formulas over a given theory. We show that admissible consequence for Σ 1 0 -substitutions over HA coincides with NNIL -preservativity over intuitionistic propositional logic . (...) Download Export citation Bookmark 17 citations
No Escape from Vardanyan's theorem.Albert Visser & Maartje de Jonge - 2006 - Archive for Mathematical Logic 45 (5):539-554.details Vardanyan's theorem states that the set of PA-valid principles of Quantified Modal Logic, QML, is complete Π0 2. We generalize this result to a wide class of theories. The crucial step in the generalization is avoiding the use of Tennenbaum's Theorem. Download Export citation Bookmark 7 citations
Undecidability in diagonalizable algebras.V. Shavrukov - 1997 - Journal of Symbolic Logic 62 (1):79-116.details If a formal theory T is able to reason about its own syntax, then the diagonalizable algebra of T is defined as its Lindenbaum sentence algebra endowed with a unary operator □ which sends a sentence φ to the sentence □φ asserting the provability of φ in T. We prove that the first order theories of diagonalizable algebras of a wide class of theories are undecidable and establish some related results. Download Export citation Bookmark 6 citations
(1 other version)Substitutions of< i> Σ_< sub> 1< sup> 0-sentences: explorations between intuitionistic propositional logic and intuitionistic arithmetic. [REVIEW]Albert Visser - 2002 - Annals of Pure and Applied Logic 114 (1):227-271.details This paper is concerned with notions of consequence. On the one hand, we study admissible consequence, specifically for substitutions of Σ 1 0 -sentences over Heyting arithmetic . On the other hand, we study preservativity relations. The notion of preservativity of sentences over a given theory is a dual of the notion of conservativity of formulas over a given theory. We show that admissible consequence for Σ 1 0 -substitutions over HA coincides with NNIL -preservativity over intuitionistic propositional logic . (...) Download Export citation Bookmark 15 citations

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