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  1. A note on uniform density in weak arithmetical theories.Duccio Pianigiani & Andrea Sorbi - 2020 - Archive for Mathematical Logic 60 (1):211-225.
    Answering a question raised by Shavrukov and Visser :569–582, 2014), we show that the lattice of \-sentences ) over any computable enumerable consistent extension T of \ is uniformly dense. We also show that for every \ and \ refer to the known hierarchies of arithmetical formulas introduced by Burr for intuitionistic arithmetic) the lattices of \-sentences over any c.e. consistent extension T of the intuitionistic version of Robinson Arithmetic \ are uniformly dense. As an immediate consequence of the proof, (...)
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  • Weak essentially undecidable theories of concatenation.Juvenal Murwanashyaka - 2022 - Archive for Mathematical Logic 61 (7):939-976.
    In the language \(\lbrace 0, 1, \circ, \preceq \rbrace \), where 0 and 1 are constant symbols, \(\circ \) is a binary function symbol and \(\preceq \) is a binary relation symbol, we formulate two theories, \( \textsf {WD} \) and \( {\textsf {D}}\), that are mutually interpretable with the theory of arithmetic \( {\textsf {R}} \) and Robinson arithmetic \({\textsf {Q}} \), respectively. The intended model of \( \textsf {WD} \) and \( {\textsf {D}}\) is the free semigroup generated (...)
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  • Finding the limit of incompleteness I.Yong Cheng - 2020 - Bulletin of Symbolic Logic 26 (3-4):268-286.
    In this paper, we examine the limit of applicability of Gödel’s first incompleteness theorem. We first define the notion “$\textsf {G1}$ holds for the theory $T$”. This paper is motivated by the following question: can we find a theory with a minimal degree of interpretation for which $\textsf {G1}$ holds. To approach this question, we first examine the following question: is there a theory T such that Robinson’s $\mathbf {R}$ interprets T but T does not interpret $\mathbf {R}$ and $\textsf (...)
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  • Current Research on Gödel’s Incompleteness Theorems.Yong Cheng - 2021 - Bulletin of Symbolic Logic 27 (2):113-167.
    We give a survey of current research on Gödel’s incompleteness theorems from the following three aspects: classifications of different proofs of Gödel’s incompleteness theorems, the limit of the applicability of Gödel’s first incompleteness theorem, and the limit of the applicability of Gödel’s second incompleteness theorem.
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