Switch to: References

Add citations

You must login to add citations.
  1. On Logical and Scientific Strength.Luca Incurvati & Carlo Nicolai - manuscript
    The notion of strength has featured prominently in recent debates about abductivism in the epistemology of logic. Following Williamson and Russell, we distinguish between logical and scientific strength and discuss the limits of the characterizations they employ. We then suggest understanding logical strength in terms of interpretability strength and scientific strength as a special case of logical strength. We present applications of the resulting notions to comparisons between logics in the traditional sense and mathematical theories.
    Direct download  
     
    Export citation  
     
    Bookmark  
  • On Positive Local Combinatorial Dividing-Lines in Model Theory.Vincent Guingona & Cameron Donnay Hill - 2019 - Archive for Mathematical Logic 58 (3-4):289-323.
    We introduce the notion of positive local combinatorial dividing-lines in model theory. We show these are equivalently characterized by indecomposable algebraically trivial Fraïssé classes and by complete prime filter classes. We exhibit the relationship between this and collapse-of-indiscernibles dividing-lines. We examine several test cases, including those arising from various classes of hypergraphs.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  • On Interpretations of Bounded Arithmetic and Bounded Set Theory.Richard Pettigrew - 2009 - Notre Dame Journal of Formal Logic 50 (2):141-152.
    In 'On interpretations of arithmetic and set theory', Kaye and Wong proved the following result, which they considered to belong to the folklore of mathematical logic.

    THEOREM 1 The first-order theories of Peano arithmetic and Zermelo-Fraenkel set theory with the axiom of infinity negated are bi-interpretable.

    In this note, I describe a theory of sets that is bi-interpretable with the theory of bounded arithmetic IDelta0 + exp. Because of the weakness of this theory of sets, I cannot straightforwardly adapt Kaye and Wong's (...)
    Direct download (9 more)  
     
    Export citation  
     
    Bookmark  
  • Finitary Set Theory.Laurence Kirby - 2009 - Notre Dame Journal of Formal Logic 50 (3):227-244.
    I argue for the use of the adjunction operator (adding a single new element to an existing set) as a basis for building a finitary set theory. It allows a simplified axiomatization for the first-order theory of hereditarily finite sets based on an induction schema and a rigorous characterization of the primitive recursive set functions. The latter leads to a primitive recursive presentation of arithmetical operations on finite sets.
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   8 citations  
  • Does Choice Really Imply Excluded Middle? Part II: Historical, Philosophical, and Foundational Reflections on the Goodman–Myhill Result†.Neil Tennant - forthcoming - Philosophia Mathematica.
    ABSTRACTOur regimentation of Goodman and Myhill’s proof of Excluded Middle revealed among its premises a form of Choice and an instance of Separation.Here we revisit Zermelo’s requirement that the separating property be definite. The instance that Goodman and Myhill used is not constructively warranted. It is that principle, and not Choice alone, that precipitates Excluded Middle.Separation in various axiomatizations of constructive set theory is examined. We conclude that insufficient critical attention has been paid to how those forms of Separation fail, (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  • Incompleteness Via Paradox and Completeness.Walter Dean - 2020 - Review of Symbolic Logic 13 (3):541-592.
    This paper explores the relationship borne by the traditional paradoxes of set theory and semantics to formal incompleteness phenomena. A central tool is the application of the Arithmetized Completeness Theorem to systems of second-order arithmetic and set theory in which various “paradoxical notions” for first-order languages can be formalized. I will first discuss the setting in which this result was originally presented by Hilbert & Bernays and also how it was later adapted by Kreisel and Wang in order to obtain (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  • 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 \. (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  • On Morita Equivalence and Interpretability.Paul Anh Mceldowney - 2020 - Review of Symbolic Logic 13 (2):388-415.
    In a recent article, Barrett & Halvorson define a notion of equivalence for first-order theories, which they call “Morita equivalence.” To argue that Morita equivalence is a reasonable measure of “theoretical equivalence,” they make use of the claim that Morita extensions “say no more” than the theories they are extending. The goal of this article is to challenge this central claim by raising objections to their argument for it and by showing why there is good reason to think that the (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  • Amphi-ZF: Axioms for Conway Games.Michael Cox & Richard Kaye - 2012 - Archive for Mathematical Logic 51 (3-4):353-371.
    A theory of two-sided containers, denoted ZF2, is introduced. This theory is then shown to be synonymous to ZF in the sense of Visser (2006), via an interpretation involving Quine pairs. Several subtheories of ZF2, and their relationships with ZF, are also examined. We include a short discussion of permutation models (in the sense of Rieger–Bernays) over ZF2. We close with highlighting some areas for future research, mostly motivated by the need to understand non-wellfounded games.
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark  
  • On the Concept of Finitism.Luca Incurvati - 2015 - Synthese 192 (8):2413-2436.
    At the most general level, the concept of finitism is typically characterized by saying that finitistic mathematics is that part of mathematics which does not appeal to completed infinite totalities and is endowed with some epistemological property that makes it secure or privileged. This paper argues that this characterization can in fact be sharpened in various ways, giving rise to different conceptions of finitism. The paper investigates these conceptions and shows that they sanction different portions of mathematics as finitistic.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  • Every Countable Model of Set Theory Embeds Into its Own Constructible Universe.Joel David Hamkins - 2013 - Journal of Mathematical Logic 13 (2):1350006.
    The main theorem of this article is that every countable model of set theory 〈M, ∈M〉, including every well-founded model, is isomorphic to a submodel of its own constructible universe 〈LM, ∈M〉 by means of an embedding j : M → LM. It follows from the proof that the countable models of set theory are linearly pre-ordered by embeddability: if 〈M, ∈M〉 and 〈N, ∈N〉 are countable models of set theory, then either M is isomorphic to a submodel of N (...)
    Direct download (8 more)  
     
    Export citation  
     
    Bookmark   6 citations  
  • Indivisible Sets and Well‐Founded Orientations of the Rado Graph.Nathanael L. Ackerman & Will Brian - 2019 - Mathematical Logic Quarterly 65 (1):46-56.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  • Substandard Models of Finite Set Theory.Laurence Kirby - 2010 - Mathematical Logic Quarterly 56 (6):631-642.
    A survey of the isomorphic submodels of Vω, the set of hereditarily finite sets. In the usual language of set theory, Vω has 2ℵ0 isomorphic submodels. But other set-theoretic languages give different systems of submodels. For example, the language of adjunction allows only countably many isomorphic submodels of Vω.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  • Truth in Generic Cuts.Richard Kaye & Tin Lok Wong - 2010 - Annals of Pure and Applied Logic 161 (8):987-1005.
    In an earlier paper the first author initiated the study of generic cuts of a model of Peano arithmetic relative to a notion of an indicator in the model. This paper extends that work. We generalise the idea of an indicator to a related neighbourhood system; this allows the theory to be extended to one that includes the case of elementary cuts. Most results transfer to this more general context, and in particular we obtain the idea of a generic cut (...)
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  • The Strength of Extensionality II—Weak Weak Set Theories Without Infinity.Kentaro Sato - 2011 - Annals of Pure and Applied Logic 162 (8):579-646.
    By obtaining several new results on Cook-style two-sorted bounded arithmetic, this paper measures the strengths of the axiom of extensionality and of other weak fundamental set-theoretic axioms in the absence of the axiom of infinity, following the author’s previous work [K. Sato, The strength of extensionality I — weak weak set theories with infinity, Annals of Pure and Applied Logic 157 234–268] which measures them in the presence. These investigations provide a uniform framework in which three different kinds of reverse (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   4 citations  
  • Rudimentary and Arithmetical Constructive Set Theory.Peter Aczel - 2013 - Annals of Pure and Applied Logic 164 (4):396-415.
    The aim of this paper is to formulate and study two weak axiom systems for the conceptual framework of constructive set theory . Arithmetical CST is just strong enough to represent the class of von Neumann natural numbers and its arithmetic so as to interpret Heyting Arithmetic. Rudimentary CST is a very weak subsystem that is just strong enough to represent a constructive version of Jensenʼs rudimentary set theoretic functions and their theory. The paper is a contribution to the study (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark  
  • Equivalences for Truth Predicates.Carlo Nicolai - 2017 - Review of Symbolic Logic 10 (2):322-356.
    One way to study and understand the notion of truth is to examine principles that we are willing to associate with truth, often because they conform to a pre-theoretical or to a semi-formal characterization of this concept. In comparing different collections of such principles, one requires formally precise notions of inter-theoretic reduction that are also adequate to compare these conceptual aspects. In this work I study possible ways to make precise the relation of conceptual equivalence between notions of truth associated (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   2 citations