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The∀∃ theory of Peano Σ1 sentences

Per Lindström & V. Yu Shavrukov

Journal of Mathematical Logic 8 (2):251-280 (2008)

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Interpretability degrees of finitely axiomatized sequential theories.Albert Visser - 2014 - Archive for Mathematical Logic 53 (1-2):23-42.details In this paper we show that the degrees of interpretability of finitely axiomatized extensions-in-the-same-language of a finitely axiomatized sequential theory—like Elementary Arithmetic EA, IΣ1, or the Gödel–Bernays theory of sets and classes GB—have suprema. This partially answers a question posed by Švejdar in his paper (Commentationes Mathematicae Universitatis Carolinae 19:789–813, 1978). The partial solution of Švejdar’s problem follows from a stronger fact: the convexity of the degree structure of finitely axiomatized extensions-in-the-same-language of a finitely axiomatized sequential theory in the degree (...) Download Export citation Bookmark 2 citations
Effectively inseparable Boolean algebras in lattices of sentences.V. Yu Shavrukov - 2010 - Archive for Mathematical Logic 49 (1):69-89.details We show the non-arithmeticity of 1st order theories of lattices of Σ n sentences modulo provable equivalence in a formal theory, of diagonalizable algebras of a wider class of arithmetic theories than has been previously known, and of the lattice of degrees of interpretability over PA. The first two results are applications of Nies’ theorem on the non-arithmeticity of the 1st order theory of the lattice of r.e. ideals on any effectively dense r.e. Boolean algebra. The theorem on degrees of (...) Download Export citation Bookmark 4 citations
Duality, non-standard elements, and dynamic properties of r.e. sets.V. Yu Shavrukov - 2016 - Annals of Pure and Applied Logic 167 (10):939-981.details Download Export citation Bookmark
Corrigendum: The ∀∃ theory of Peano Σ1 sentences.V. Yu Shavrukov - 2017 - Journal of Mathematical Logic 17 (1):1792001.details Download Export citation Bookmark
On Shavrukov’s Non-Isomorphism Theorem for Diagonalizable Algebras.Evgeny A. Kolmakov - 2024 - Review of Symbolic Logic 17 (1):206-243.details We prove a strengthened version of Shavrukov’s result on the non-isomorphism of diagonalizable algebras of two $\Sigma _1$ -sound theories, based on the improvements previously found by Adamsson. We then obtain several corollaries to the strengthened result by applying it to various pairs of theories and obtain new non-isomorphism examples. In particular, we show that there are no surjective homomorphisms from the algebra $(\mathfrak {L}_T, \Box _T\Box _T)$ onto the algebra $(\mathfrak {L}_T, \Box _T)$. The case of bimodal diagonalizable algebras (...) Download Export citation Bookmark
Hierarchical Incompleteness Results for Arithmetically Definable Extensions of Fragments of Arithmetic.Rasmus Blanck - 2021 - Review of Symbolic Logic 14 (3):624-644.details There has been a recent interest in hierarchical generalizations of classic incompleteness results. This paper provides evidence that such generalizations are readily obtainable from suitably formulated hierarchical versions of the principles used in the original proofs. By collecting such principles, we prove hierarchical versions of Mostowski’s theorem on independent formulae, Kripke’s theorem on flexible formulae, Woodin’s theorem on the universal algorithm, and a few related results. As a corollary, we obtain the expected result that the formula expressing “$\mathrm {T}$is$\Sigma _n$-ill” (...) Download Export citation Bookmark 1 citation
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

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