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  1. Essential hereditary undecidability.Albert Visser - forthcoming - Archive for Mathematical Logic:1-34.
    In this paper we study essential hereditary undecidability. Theories with this property are a convenient tool to prove undecidability of other theories. The paper develops the basic facts concerning essentially hereditary undecidability and provides salient examples, like a construction of essentially hereditarily undecidable theories due to Hanf and an example of a rather natural essentially hereditarily undecidable theory strictly below. We discuss the (non-)interaction of essential hereditary undecidability with recursive boolean isomorphism. We develop a reduction relation essential tolerance, or, in (...)
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  • Effectively inseparable Boolean algebras in lattices of sentences.V. Yu Shavrukov - 2010 - Archive for Mathematical Logic 49 (1):69-89.
    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 (...)
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  • A Recursion‐theoretic View of Axiomatizable Theories.Marian Boykan Pour-El - 1970 - Dialectica 24 (4):267-276.
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  • Rekursive Untrennbarkeit Bei Elementaren Theorien.Hans-Dietrich Hecker - 1971 - Mathematical Logic Quarterly 17 (1):443-463.
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  • The Boolean algebra of formulas of first-order logic.Don H. Faust - 1982 - Annals of Mathematical Logic 23 (1):27.
    The algebraic recursive structure of countable languages of classical first-order logic with equality is analysed. all languages of finite undecidable similarity type are shown to be algebraically and recursively equivalent in the following sense: their boolean algebras of formulas are, after trivial factors involving the one element models of the languages have been excepted, recursively isomorphic by a map which preserves the degree of recursiveness of their models.
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  • There Are No Minimal Effectively Inseparable Theories.Yong Cheng - 2023 - Notre Dame Journal of Formal Logic 64 (4):425-439.
    This paper belongs to the research on the limit of the first incompleteness theorem. Effectively inseparable (EI) theories can be viewed as an effective version of essentially undecidable (EU) theories, and EI is stronger than EU. We examine this question: Are there minimal effectively inseparable theories with respect to interpretability? We propose tEI, the theory version of EI. We first prove that there are no minimal tEI theories with respect to interpretability (i.e., for any tEI theory T, we can effectively (...)
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