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  1. Passive induction and a solution to a Paris–Wilkie open question.Dan E. Willard - 2007 - Annals of Pure and Applied Logic 146 (2-3):124-149.
    In 1981, Paris and Wilkie raised the open question about whether and to what extent the axiom system did satisfy the Second Incompleteness Theorem under Semantic Tableaux deduction. Our prior work showed that the semantic tableaux version of the Second Incompleteness Theorem did generalize for the most common definition of appearing in the standard textbooks.However, there was an alternate interesting definition of this axiom system in the Wilkie–Paris article in the Annals of Pure and Applied Logic 35 , pp. 261–302 (...)
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  • Boolos-style proofs of limitative theorems.György Serény - 2004 - Mathematical Logic Quarterly 50 (2):211.
    Boolos's proof of incompleteness is extended straightforwardly to yield simple “diagonalization-free” proofs of some classical limitative theorems of logic.
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  • The incompleteness theorems after 70 years.Henryk Kotlarski - 2004 - Annals of Pure and Applied Logic 126 (1-3):125-138.
    We give some information about new proofs of the incompleteness theorems, found in 1990s. Some of them do not require the diagonal lemma as a method of construction of an independent statement.
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  • Semi-honest subrecursive degrees and the collection rule in arithmetic.Andrés Cordón-Franco & F. Félix Lara-Martín - 2023 - Archive for Mathematical Logic 63 (1):163-180.
    By a result of L.D. Beklemishev, the hierarchy of nested applications of the $$\Sigma _1$$ -collection rule over any $$\Pi _2$$ -axiomatizable base theory extending Elementary Arithmetic collapses to its first level. We prove that this result cannot in general be extended to base theories of arbitrary quantifier complexity. In fact, given any recursively enumerable set of true $$\Pi _2$$ -sentences, S, we construct a sound $$(\Sigma _2 \! \vee \! \Pi _2)$$ -axiomatized theory T extending S such that the (...)
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  • Löb's theorem in a set theoretical setting.Cezary Cieśliński - 2003 - Studia Logica 75 (3):319 - 326.
    We present a semantic proof of Löb's theorem for theories T containing ZF. Without using the diagonalization lemma, we construct a sentence AUT T, which says intuitively that the predicate autological with respect to T (i.e. applying to itself in every model of T) is itself autological with respect to T. In effect, the sentence AUT T states I follow semantically from T. Then we show that this sentence indeed follows from T and therefore is true.
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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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  • The Scope of Gödel’s First Incompleteness Theorem.Bernd Buldt - 2014 - Logica Universalis 8 (3-4):499-552.
    Guided by questions of scope, this paper provides an overview of what is known about both the scope and, consequently, the limits of Gödel’s famous first incompleteness theorem.
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  • Preservation theorems and restricted consistency statements in bounded arithmetic.Arnold Beckmann - 2004 - Annals of Pure and Applied Logic 126 (1-3):255-280.
    We define and study a new restricted consistency notion RCon ∗ for bounded arithmetic theories T 2 j . It is the strongest ∀ Π 1 b -statement over S 2 1 provable in T 2 j , similar to Con in Krajíček and Pudlák, 29) or RCon in Krajı́ček and Takeuti 107). The advantage of our notion over the others is that RCon ∗ can directly be used to construct models of T 2 j . We apply this by (...)
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  • Partial collapses of the complexity hierarchy in models for fragments of bounded arithmetic.Zofia Adamowicz & Leszek Aleksander Kołodziejczyk - 2007 - Annals of Pure and Applied Logic 145 (1):91-95.
    For any n, we construct a model of in which each formula is equivalent to an formula.
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