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  1. Growing Commas. A Study of Sequentiality and Concatenation.Albert Visser - 2009 - Notre Dame Journal of Formal Logic 50 (1):61-85.
    In his paper "Undecidability without arithmetization," Andrzej Grzegorczyk introduces a theory of concatenation $\mathsf{TC}$. We show that pairing is not definable in $\mathsf{TC}$. We determine a reasonable extension of $\mathsf{TC}$ that is sequential, that is, has a good sequence coding.
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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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  • Bases for Structures and Theories I.Jeffrey Ketland - 2020 - Logica Universalis 14 (3):357-381.
    Sometimes structures or theories are formulated with different sets of primitives and yet are definitionally equivalent. In a sense, the transformations between such equivalent formulations are rather like basis transformations in linear algebra or co-ordinate transformations in geometry. Here an analogous idea is investigated. Let a relational signature \ be given. For a set \ of \-formulas, we introduce a corresponding set \ of new relation symbols and a set of explicit definitions of the \ in terms of the \. (...)
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  • Deflationism beyond arithmetic.Kentaro Fujimoto - 2019 - Synthese 196 (3):1045-1069.
    The conservativeness argument poses a dilemma to deflationism about truth, according to which a deflationist theory of truth must be conservative but no adequate theory of truth is conservative. The debate on the conservativeness argument has so far been framed in a specific formal setting, where theories of truth are formulated over arithmetical base theories. I will argue that the appropriate formal setting for evaluating the conservativeness argument is provided not by theories of truth over arithmetic but by those over (...)
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  • Peano Corto and Peano Basso: A Study of Local Induction in the Context of Weak Theories.Albert Visser - 2014 - Mathematical Logic Quarterly 60 (1-2):92-117.
    In this paper we study local induction w.r.t. Σ1‐formulas over the weak arithmetic. The local induction scheme, which was introduced in, says roughly this: for any virtual class that is progressive, i.e., is closed under zero and successor, and for any non‐empty virtual class that is definable by a Σ1‐formula without parameters, the intersection of and is non‐empty. In other words, we have, for all Σ1‐sentences S, that S implies, whenever is progressive. Since, in the weak context, we have (at (...)
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  • Stable and Unstable Theories of Truth and Syntax.Beau Madison Mount & Daniel Waxman - 2021 - Mind 130 (518):439-473.
    Recent work on formal theories of truth has revived an approach, due originally to Tarski, on which syntax and truth theories are sharply distinguished—‘disentangled’—from mathematical base theories. In this paper, we defend a novel philosophical constraint on disentangled theories. We argue that these theories must be epistemically stable: they must possess an intrinsic motivation justifying no strictly stronger theory. In a disentangled setting, even if the base and the syntax theory are individually stable, they may be jointly unstable. We contend (...)
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  • Against Harmony: Infinite Idealizations and Causal Explanation.Iulian D. Toader - 2015 - In Ilie Parvu, Gabriel Sandu & Iulian D. Toader (eds.), Romanian Studies in Philosophy of Science. Boston Studies in the Philosophy and History of Science, vol. 313: Springer. pp. 291-301.
    This paper argues against the view that the standard explanation of phase transitions in statistical mechanics may be considered a causal explanation, a distortion that can nevertheless successfully represent causal relations.
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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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  • Weak theories of concatenation and minimal essentially undecidable theories: An encounter of WTC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf{WTC}}$$\end{document} and S2S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf{S2S}}$$\end{document}.Kojiro Higuchi & Yoshihiro Horihata - 2014 - Archive for Mathematical Logic 53 (7-8):835-853.
    We consider weak theories of concatenation, that is, theories for strings or texts. We prove that the theory of concatenation WTC-ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf{WTC}^{-\varepsilon}}$$\end{document}, which is a weak subtheory of Grzegorczyk’s theory TC-ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf{TC}^{-\varepsilon}}$$\end{document}, is a minimal essentially undecidable theory, that is, the theory WTC-ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf{WTC}^{-\varepsilon}}$$\end{document} is essentially undecidable and if one omits an axiom scheme from WTC-ε\documentclass[12pt]{minimal} \usepackage{amsmath} (...)
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  • Arithmetic on semigroups.Mihai Ganea - 2009 - Journal of Symbolic Logic 74 (1):265-278.
    Relations between some theories of semigroups (also known as theories of strings or theories of concatenation) and arithmetic are surveyed. In particular Robinson's arithmetic Q is shown to be mutually interpretable with TC, a weak theory of concatenation introduced by Grzegorczyk. Furthermore, TC is shown to be interpretable in the theory F studied by Tarski and Szmielewa, thus confirming their claim that F is essentially undecidable.
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  • First-order concatenation theory with bounded quantifiers.Lars Kristiansen & Juvenal Murwanashyaka - 2020 - Archive for Mathematical Logic 60 (1):77-104.
    We study first-order concatenation theory with bounded quantifiers. We give axiomatizations with interesting properties, and we prove some normal-form results. Finally, we prove a number of decidability and undecidability results.
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  • Weak Theories of Concatenation and Arithmetic.Yoshihiro Horihata - 2012 - Notre Dame Journal of Formal Logic 53 (2):203-222.
    We define a new theory of concatenation WTC which is much weaker than Grzegorczyk's well-known theory TC. We prove that WTC is mutually interpretable with the weak theory of arithmetic R. The latter is, in a technical sense, much weaker than Robinson's arithmetic Q, but still essentially undecidable. Hence, as a corollary, WTC is also essentially undecidable.
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  • Towards metamathematics of weak arithmetics over fuzzy logic.Petr Hájek - 2011 - Logic Journal of the IGPL 19 (3):467-475.
    This paper continues investigation of a very weak arithmetic FQ∼ that results from the well-known Robinson arithmetic Q by not assuming that addition and multiplication are total functions and, secondly, by weakening the classical logic to the basic mathematical fuzzy logic BL∀ . This investigation was started in the paper [5] where the first Gödel incompleteness of FQ∼ is proved. Here we first discuss Q∼ over the Gödel fuzzy logic G∀, or alternatively over the intuitionistic predicate logic, showing essential incompleteness (...)
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  • Mutual interpretability of Robinson arithmetic and adjunctive set theory with extensionality.Zlatan Damnjanovic - 2017 - Bulletin of Symbolic Logic 23 (4):381-404.
    An elementary theory of concatenation,QT+, is introduced and used to establish mutual interpretability of Robinson arithmetic, Minimal Predicative Set Theory, quantifier-free part of Kirby’s finitary set theory, and Adjunctive Set Theory, with or without extensionality. The most basic arithmetic and simplest set theory thus turn out to be variants of string theory.
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