Results for 'omega-categoricity'

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  1.  3
    Omega-Categoricity, Relative Categoricity and Coordinatisation.Wilfrid Hodges, I. M. Hodkinson & Dugald Macpherson - 1990 - Annals of Pure and Applied Logic 46 (2):169-199.
  2.  28
    Categoricity of Theories in "L" Kappa Omega with Kappa a Compact Cardinal.S. Shelah - 1990 - Annals of Pure and Applied Logic 47 (1):41.
  3.  21
    A Note on Generic Projective Planes.Koichiro Ikeda - 2002 - Notre Dame Journal of Formal Logic 43 (4):249-254.
    Hrushovski constructed an -categorical stable pseudoplane which refuted Lachlan's conjecture. In this note, we show that an -categorical projective plane cannot be constructed by "the Hrushovski method.".
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  4. CORCORAN'S 27 ENTRIES IN THE 1999 SECOND EDITION.John Corcoran - 1999 - In Robert Audi (ed.), The Cambridge Dictionary of Philosophy. CAMBRIDGE UP. pp. 65-941.
    Corcoran’s 27 entries in the 1999 second edition of Robert Audi’s Cambridge Dictionary of Philosophy [Cambridge: Cambridge UP]. -/- ancestral, axiomatic method, borderline case, categoricity, Church (Alonzo), conditional, convention T, converse (outer and inner), corresponding conditional, degenerate case, domain, De Morgan, ellipsis, laws of thought, limiting case, logical form, logical subject, material adequacy, mathematical analysis, omega, proof by recursion, recursive function theory, scheme, scope, Tarski (Alfred), tautology, universe of discourse. -/- The entire work is available online free at more than (...)
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  5.  39
    Categoricity of Computable Infinitary Theories.W. Calvert, S. S. Goncharov, J. F. Knight & Jessica Millar - 2009 - Archive for Mathematical Logic 48 (1):25-38.
    Computable structures of Scott rank ${\omega_1^{CK}}$ are an important boundary case for structural complexity. While every countable structure is determined, up to isomorphism, by a sentence of ${\mathcal{L}_{\omega_1 \omega}}$ , this sentence may not be computable. We give examples, in several familiar classes of structures, of computable structures with Scott rank ${\omega_1^{CK}}$ whose computable infinitary theories are each ${\aleph_0}$ -categorical. General conditions are given, covering many known methods for constructing computable structures with Scott rank ${\omega_1^{CK}}$ , which guarantee that the (...)
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  6.  14
    Categoricity and Possibility. A Note on Williamson's Modal Monism.Iulian D. Toader - forthcoming - In Martin Blicha & Igor Sedlár (eds.), The Logica Yearbook 2019. London: College Publications. pp. 1-11.
    The paper sketches an argument against modal monism, more specifically against the reduction of physical possibility to metaphysical possibility. The argument is based on the non-categoricity of quantum logic.
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  7.  13
    Categoricity Spectra for Rigid Structures.Ekaterina Fokina, Andrey Frolov & Iskander Kalimullin - 2016 - Notre Dame Journal of Formal Logic 57 (1):45-57.
    For a computable structure $\mathcal {M}$, the categoricity spectrum is the set of all Turing degrees capable of computing isomorphisms among arbitrary computable copies of $\mathcal {M}$. If the spectrum has a least degree, this degree is called the degree of categoricity of $\mathcal {M}$. In this paper we investigate spectra of categoricity for computable rigid structures. In particular, we give examples of rigid structures without degrees of categoricity.
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  8. Categoricity.John Corcoran - 1980 - History and Philosophy of Logic 1 (1):187-207.
    After a short preface, the first of the three sections of this paper is devoted to historical and philosophic aspects of categoricity. The second section is a self-contained exposition, including detailed definitions, of a proof that every mathematical system whose domain is the closure of its set of distinguished individuals under its distinguished functions is categorically characterized by its induction principle together with its true atoms (atomic sentences and negations of atomic sentences). The third section deals with applications especially those (...)
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  9.  93
    Relative Categoricity and Abstraction Principles.Sean Walsh & Sean Ebels-Duggan - 2015 - Review of Symbolic Logic 8 (3):572-606.
    Many recent writers in the philosophy of mathematics have put great weight on the relative categoricity of the traditional axiomatizations of our foundational theories of arithmetic and set theory. Another great enterprise in contemporary philosophy of mathematics has been Wright's and Hale's project of founding mathematics on abstraction principles. In earlier work, it was noted that one traditional abstraction principle, namely Hume's Principle, had a certain relative categoricity property, which here we term natural relative categoricity. In this paper, we show (...)
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  10. Structure and Categoricity: Determinacy of Reference and Truth Value in the Philosophy of Mathematics.Tim Button & Sean Walsh - 2016 - Philosophia Mathematica 24 (3):283-307.
    This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity arguments are sufficient to secure reference to mathematical structures up to isomorphism, we assess what exactly is achieved by recent ‘internal’ renditions of the famous categoricity arguments for arithmetic and set theory.
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  11.  47
    Degrees of Categoricity of Computable Structures.Ekaterina B. Fokina, Iskander Kalimullin & Russell Miller - 2010 - Archive for Mathematical Logic 49 (1):51-67.
    Defining the degree of categoricity of a computable structure ${\mathcal{M}}$ to be the least degree d for which ${\mathcal{M}}$ is d-computably categorical, we investigate which Turing degrees can be realized as degrees of categoricity. We show that for all n, degrees d.c.e. in and above 0 (n) can be so realized, as can the degree 0 (ω).
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  12.  41
    Speech Acts, Categoricity, and the Meanings of Logical Connectives.Ole Thomassen Hjortland - 2014 - Notre Dame Journal of Formal Logic 55 (4):445-467.
    In bilateral systems for classical logic, assertion and denial occur as primitive signs on formulas. Such systems lend themselves to an inferentialist story about how truth-conditional content of connectives can be determined by inference rules. In particular, for classical logic there is a bilateral proof system which has a property that Carnap in 1943 called categoricity. We show that categorical systems can be given for any finite many-valued logic using $n$-sided sequent calculus. These systems are understood as a further development (...)
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  13.  27
    Internal Categoricity in Arithmetic and Set Theory.Jouko Väänänen & Tong Wang - 2015 - Notre Dame Journal of Formal Logic 56 (1):121-134.
    We show that the categoricity of second-order Peano axioms can be proved from the comprehension axioms. We also show that the categoricity of second-order Zermelo–Fraenkel axioms, given the order type of the ordinals, can be proved from the comprehension axioms. Thus these well-known categoricity results do not need the so-called “full” second-order logic, the Henkin second-order logic is enough. We also address the question of “consistency” of these axiom systems in the second-order sense, that is, the question of existence of (...)
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  14.  11
    Degrees That Are Not Degrees of Categoricity.Bernard Anderson & Barbara Csima - 2016 - Notre Dame Journal of Formal Logic 57 (3):389-398.
    A computable structure $\mathcal {A}$ is $\mathbf {x}$-computably categorical for some Turing degree $\mathbf {x}$ if for every computable structure $\mathcal {B}\cong\mathcal {A}$ there is an isomorphism $f:\mathcal {B}\to\mathcal {A}$ with $f\leq_{T}\mathbf {x}$. A degree $\mathbf {x}$ is a degree of categoricity if there is a computable structure $\mathcal {A}$ such that $\mathcal {A}$ is $\mathbf {x}$-computably categorical, and for all $\mathbf {y}$, if $\mathcal {A}$ is $\mathbf {y}$-computably categorical, then $\mathbf {x}\leq_{T}\mathbf {y}$. We construct a $\Sigma^{0}_{2}$ set whose degree (...)
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  15.  6
    On the Complexity of Categoricity in Computable Structures.Walker M. White - 2003 - Mathematical Logic Quarterly 49 (6):603.
    We investigate the computational complexity the class of Γ-categorical computable structures. We show that hyperarithmetic categoricity is Π11-complete, while computable categoricity is Π04-hard.
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  16.  18
    Categoricity Spectra for Polymodal Algebras.Nikolay Bazhenov - 2016 - Studia Logica 104 (6):1083-1097.
    We investigate effective categoricity for polymodal algebras. We prove that the class of polymodal algebras is complete with respect to degree spectra of nontrivial structures, effective dimensions, expansion by constants, and degree spectra of relations. In particular, this implies that every categoricity spectrum is the categoricity spectrum of a polymodal algebra.
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  17. Categoricity Theorems and Conceptions of Set.Gabriel Uzquiano - 2002 - Journal of Philosophical Logic 31 (2):181-196.
    Two models of second-order ZFC need not be isomorphic to each other, but at least one is isomorphic to an initial segment of the other. The situation is subtler for impure set theory, but Vann McGee has recently proved a categoricity result for second-order ZFCU plus the axiom that the urelements form a set. Two models of this theory with the same universe of discourse need not be isomorphic to each other, but the pure sets of one are isomorphic to (...)
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  18.  18
    On Absoluteness of Categoricity in Abstract Elementary Classes.Sy-David Friedman & Martin Koerwien - 2011 - Notre Dame Journal of Formal Logic 52 (4):395-402.
    Shelah has shown that $\aleph_1$-categoricity for Abstract Elementary Classes (AECs) is not absolute in the following sense: There is an example $K$ of an AEC (which is actually axiomatizable in the logic $L(Q)$) such that if $2^{\aleph_0}.
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  19.  4
    Integrating TPS and OMEGA.Christoph Benzmüller, Matt Bishop & Volker Sorge - 1999 - Journal of Universal Computer Science 5 (3):188-207.
    This paper reports on the integration of the higher-order theorem proving environment TPS [Andrews96] into the mathematical assistant OMEGA [Omega97]. TPS can be called from OMEGA either as a black box or as an interactive system. In black box mode, the user has control over the parameters which control proof search in TPS; in interactive mode, all features of the TPS-system are available to the user. If the subproblem which is passed to TPS contains concepts defined in OMEGA’s database of (...)
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  20.  40
    On an Application of Categoricity.Alexander Paseau - 2005 - Proceedings of the Aristotelian Society 105 (3):411–415.
    James Walmsley in “Categoricity and Indefinite Extensibility” argues that a realist about some branch of mathematics X (e.g. arithmetic) apparently cannot use the categoricity of an axiomatisation of X to justify her belief that every sentence of the language of X has a truth-value. My note corrects Walmsley’s formulation of his claim, and shows that his argument for it hinges on the implausible idea that grasping that there is some model of the axioms amounts to grasping that there is a (...)
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  21.  7
    Categoricity and Mathematical Knowledge.Fernando Ferreira - 2017 - Revista Portuguesa de Filosofia 73 (3-4):1423-1436.
    We argue that the basic notions of mathematics can only be properly formulated in an informal way. Mathematical notions transcend formalizations and their study involves the consideration of other mathematical notions. We explain the fundamental role of categoricity theorems in making these studies possible. We arrive at the conclusion that the enterprise of mathematics is not infallible and that it ultimately relies on degrees of evidence.
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  22.  8
    Ultrafilters of Character $Omega_1$.Klaas Pieter Hart - 1989 - Journal of Symbolic Logic 54 (1):1-15.
    Using side-by-side Sacks forcing, it is shown that it is consistent that $2^\omega$ be large and that there be many types of ultrafilters of character $\omega_1$.
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  23. Categoricity and Negation. A Note on Kripke’s Affirmativism.Constantin C. Brîncuș & Iulian D. Toader - 2018 - In The Logica Yearbook 2018. London: College Publications. pp. 57-66.
    The idea that an adequate language for science needs a negation operator was recently dismissed by Kripke as "yet another dogma of empiricism". That a scientist could, and even should, drop negation implies at least three points: 1. negativist theories, i.e., theories formulated in languages that include negation, are conservative extensions of their affirmativist versions; 2. negativist theories have no serious advantages over their affirmativist versions; 3. negativist theories are dispensable and should better be replaced by their affirmativist versions. We (...)
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  24. Inferentialism and the Categoricity Problem: Reply to Raatikainen.Julien Murzi & Ole Thomassen Hjortland - 2009 - Analysis 69 (3):480-488.
    It is sometimes held that rules of inference determine the meaning of the logical constants: the meaning of, say, conjunction is fully determined by either its introduction or its elimination rules, or both; similarly for the other connectives. In a recent paper, Panu Raatikainen (2008) argues that this view - call it logical inferentialism - is undermined by some "very little known" considerations by Carnap (1943) to the effect that "in a definite sense, it is not true that the standard (...)
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  25. Failures of Categoricity and Compositionality for Intuitionistic Disjunction.Jack Woods - 2012 - Thought: A Journal of Philosophy 1 (4):281-291.
    I show that the model-theoretic meaning that can be read off the natural deduction rules for disjunction fails to have certain desirable properties. I use this result to argue against a modest form of inferentialism which uses natural deduction rules to fix model-theoretic truth-conditions for logical connectives.
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  26.  75
    Completeness and Categoricity, Part I: 19th Century Axiomatics to 20th Century Metalogic.Steve Awodey & Erich H. Reck - unknown
    This paper is the first in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins in the development of the axiomatic method. We argue that, both from historical and logical points of view, higher-order logic is an appropriate framework for considering such notions, and we consider some open questions in higher-order axiomatics. In addition, we indicate how one can fruitfully extend the usual set-theoretic semantics (...)
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  27.  5
    Organisation, Transformation, and Propagation of Mathematical Knowledge in Omega.Serge Autexier, Christoph Benzmüller, Dominik Dietrich & Marc Wagner - 2008 - Mathematics in Computer Science 2 (2):253-277.
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  28.  3
    The Curious Inference of Boolos in MIZAR and OMEGA.Christoph Benzmüller & Chad Brown - 2007 - In Roman Matuszewski & Anna Zalewska (eds.), From Insight to Proof -- Festschrift in Honour of Andrzej Trybulec. The University of Bialystok, Polen. pp. 299-388.
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  29.  5
    Omega.Christoph Benzmüller, Armin Fiedler, Andreas Meier, Martin Pollet & Jörg Siekmann - 2006 - In Freek Wiedijk (ed.), The Seventeen Provers of the World. Springer. pp. 127-141.
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  30.  3
    Categoricity in Abstract Elementary Classes with No Maximal Models.Monica VanDieren - 2006 - Annals of Pure and Applied Logic 141 (1):108-147.
    The results in this paper are in a context of abstract elementary classes identified by Shelah and Villaveces in which the amalgamation property is not assumed. The long-term goal is to solve Shelah’s Categoricity Conjecture in this context. Here we tackle a problem of Shelah and Villaveces by proving that in their context, the uniqueness of limit models follows from categoricity under the assumption that the subclass of amalgamation bases is closed under unions of bounded, -increasing chains.
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  31.  10
    Omega-Inconsistency Without Cuts and Nonstandard Models.Andreas Fjellstad - 2016 - Australasian Journal of Logic 13 (5).
    This paper concerns the relationship between transitivity of entailment, omega-inconsistency and nonstandard models of arithmetic. First, it provides a cut-free sequent calculus for non-transitive logic of truth STT based on Robinson Arithmetic and shows that this logic is omega-inconsistent. It then identifies the conditions in McGee for an omega-inconsistent logic as quantified standard deontic logic, presents a cut-free labelled sequent calculus for quantified standard deontic logic based on Robinson Arithmetic where the deontic modality is treated as a predicate, proves omega-inconsistency (...)
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  32. Degrees of Categoricity and the Hyperarithmetic Hierarchy.Barbara F. Csima, Johanna N. Y. Franklin & Richard A. Shore - 2013 - Notre Dame Journal of Formal Logic 54 (2):215-231.
    We study arithmetic and hyperarithmetic degrees of categoricity. We extend a result of E. Fokina, I. Kalimullin, and R. Miller to show that for every computable ordinal $\alpha$, $\mathbf{0}^{}$ is the degree of categoricity of some computable structure $\mathcal{A}$. We show additionally that for $\alpha$ a computable successor ordinal, every degree $2$-c.e. in and above $\mathbf{0}^{}$ is a degree of categoricity. We further prove that every degree of categoricity is hyperarithmetic and show that the index set of structures with degrees (...)
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  33.  20
    Shelah's Categoricity Conjecture From a Successor for Tame Abstract Elementary Classes.Rami Grossberg & Monica Vandieren - 2006 - Journal of Symbolic Logic 71 (2):553 - 568.
    We prove a categoricity transfer theorem for tame abstract elementary classes. Theorem 0.1. Suppose that K is a χ-tame abstract elementary class and satisfies the amalgamation and joint embedding properties and has arbitrarily large models. Let λ ≥ Max{χ.LS(K)⁺}. If K is categorical in λ and λ⁺, then K is categorical in λ⁺⁺. Combining this theorem with some results from [37], we derive a form of Shelah's Categoricity Conjecture for tame abstract elementary classes: Corollary 0.2. Suppose K is a χ-tame (...)
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  34.  11
    Categoricity From One Successor Cardinal in Tame Abstract Elementary Classes.Rami Grossberg & Monica Vandieren - 2006 - Journal of Mathematical Logic 6 (2):181-201.
    We prove that from categoricity in λ+ we can get categoricity in all cardinals ≥ λ+ in a χ-tame abstract elementary classe [Formula: see text] which has arbitrarily large models and satisfies the amalgamation and joint embedding properties, provided [Formula: see text] and λ ≥ χ. For the missing case when [Formula: see text], we prove that [Formula: see text] is totally categorical provided that [Formula: see text] is categorical in [Formula: see text] and [Formula: see text].
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  35.  18
    Effective Categoricity of Equivalence Structures.Wesley Calvert, Douglas Cenzer, Valentina Harizanov & Andrei Morozov - 2006 - Annals of Pure and Applied Logic 141 (1):61-78.
    We investigate effective categoricity of computable equivalence structures . We show that is computably categorical if and only if has only finitely many finite equivalence classes, or has only finitely many infinite classes, bounded character, and at most one finite k such that there are infinitely many classes of size k. We also prove that all computably categorical structures are relatively computably categorical, that is, have computably enumerable Scott families of existential formulas. Since all computable equivalence structures are relatively categorical, (...)
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  36.  19
    Degrees of Bi-Embeddable Categoricity of Equivalence Structures.Nikolay Bazhenov, Ekaterina Fokina, Dino Rossegger & Luca San Mauro - 2019 - Archive for Mathematical Logic 58 (5-6):543-563.
    We study the algorithmic complexity of embeddings between bi-embeddable equivalence structures. We define the notions of computable bi-embeddable categoricity, \ bi-embeddable categoricity, and degrees of bi-embeddable categoricity. These notions mirror the classical notions used to study the complexity of isomorphisms between structures. We show that the notions of \ bi-embeddable categoricity and relative \ bi-embeddable categoricity coincide for equivalence structures for \. We also prove that computable equivalence structures have degree of bi-embeddable categoricity \, or \. We furthermore obtain results (...)
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  37.  27
    Categoricity Transfer in Simple Finitary Abstract Elementary Classes.Tapani Hyttinen & Meeri Kesälä - 2011 - Journal of Symbolic Logic 76 (3):759 - 806.
    We continue our study of finitary abstract elementary classes, defined in [7]. In this paper, we prove a categoricity transfer theorem for a case of simple finitary AECs. We introduce the concepts of weak κ-categoricity and f-primary models to the framework of א₀-stable simple finitary AECs with the extension property, whereby we gain the following theorem: Let (, ≼ ) be a simple finitary AEC, weakly categorical in some uncountable κ. Then (, ≼ ) is weakly categorical in (...)
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  38.  73
    Categoricity in Homogeneous Complete Metric Spaces.Åsa Hirvonen & Tapani Hyttinen - 2009 - Archive for Mathematical Logic 48 (3-4):269-322.
    We introduce a new approach to the model theory of metric structures by defining the notion of a metric abstract elementary class (MAEC) closely resembling the notion of an abstract elementary class. Further we define the framework of a homogeneous MAEC were we additionally assume the existence of arbitrarily large models, joint embedding, amalgamation, homogeneity and a property which we call the perturbation property. We also assume that the Löwenheim-Skolem number, which in this setting refers to the density character of (...)
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  39.  64
    WHAT CAN A CATEGORICITY THEOREM TELL US?Toby Meadows - 2013 - Review of Symbolic Logic (3):524-544.
    f The purpose of this paper is to investigate categoricity arguments conducted in second order logic and the philosophical conclusions that can be drawn from them. We provide a way of seeing this result, so to speak, through a first order lens divested of its second order garb. Our purpose is to draw into sharper relief exactly what is involved in this kind of categoricity proof and to highlight the fact that we should be reserved before drawing powerful philosophical conclusions (...)
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  40.  58
    Cosmology From Alpha to Omega: Response to Reviews.Robert John Russell - 2010 - Zygon 45 (1):237-250.
    I gratefully acknowledge and respond here to four reviews of my recent book, Cosmology from Alpha to Omega. Nancey Murphy stresses the importance of showing consistency between Christian theology and natural science through a detailed examination of my recent model of their creative interaction. She suggests how this model can be enhanced by adopting Alasdair MacIntyre's understanding of tradition in order to adjudicate between competing ways of incorporating science into a wider worldview. She urges the inclusion of ethics in my (...)
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  41.  7
    Δ20-Categoricity in Boolean Algebras and Linear Orderings.Charles F. D. McCoy - 2003 - Annals of Pure and Applied Logic 119 (1-3):85-120.
    We characterize Δ20-categoricity in Boolean algebras and linear orderings under some extra effectiveness conditions. We begin with a study of the relativized notion in these structures.
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  42.  17
    PFA and Ideals on $\Omega_{2}$ Whose Associated Forcings Are Proper.Sean Cox - 2012 - Notre Dame Journal of Formal Logic 53 (3):397-412.
    Given an ideal $I$ , let $\mathbb{P}_{I}$ denote the forcing with $I$ -positive sets. We consider models of forcing axioms $MA(\Gamma)$ which also have a normal ideal $I$ with completeness $\omega_{2}$ such that $\mathbb{P}_{I}\in \Gamma$ . Using a bit more than a superhuge cardinal, we produce a model of PFA (proper forcing axiom) which has many ideals on $\omega_{2}$ whose associated forcings are proper; a similar phenomenon is also observed in the standard model of $MA^{+\omega_{1}}(\sigma\mbox{-closed})$ obtained from a supercompact cardinal. (...)
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  43.  78
    Completeness and Categoricity: Frege, Gödel and Model Theory.Stephen Read - 1997 - History and Philosophy of Logic 18 (2):79-93.
    Frege?s project has been characterized as an attempt to formulate a complete system of logic adequate to characterize mathematical theories such as arithmetic and set theory. As such, it was seen to fail by Gödel?s incompleteness theorem of 1931. It is argued, however, that this is to impose a later interpretation on the word ?complete? it is clear from Dedekind?s writings that at least as good as interpretation of completeness is categoricity. Whereas few interesting first-order mathematical theories are categorical or (...)
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  44.  17
    Completeness and Categoricity : Formalization Without Foundationalism.John T. Baldwin - 2014 - Bulletin of Symbolic Logic 20 (1):39-79.
    We propose a criterion to regard a property of a theory as virtuous: the property must have significant mathematical consequences for the theory. We then rehearse results of Ajtai, Marek, Magidor, H. Friedman and Solovay to argue that for second order logic, ‘categoricity’ has little virtue. For first order logic, categoricity is trivial; but ‘categoricity in power’ has enormous structural consequences for any of the theories satisfying it. The stability hierarchy extends this virtue to other complete theories. The interaction of (...)
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  45.  10
    A Diller-Nahm-Style Functional Interpretation of $\Hbox{\Sf KP} \Omega$.Wolfgang Burr - 2000 - Archive for Mathematical Logic 39 (8):599-604.
    The Dialectica-style functional interpretation of Kripke-Platek set theory with infinity ( $\hbox{\sf KP} \omega$ ) given in [1] uses a choice functional (which is not a definable set function of ( $hbox{\sf KP} \omega$ ). By means of a Diller-Nahm-style interpretation (cf. [4]) it is possible to eliminate the choice functional and give an interpretation by set functionals primitive recursive in $x\mapsto\omega$ . This yields the following characterization: The class of $\Sigma$ -definable set functions of $\hbox{\sf KP} \omega$ coincides with (...)
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  46.  18
    Domains of Sciences, Universes of Discourse and Omega Arguments.Jose M. Saguillo - 1999 - History and Philosophy of Logic 20 (3-4):267-290.
    Each science has its own domain of investigation, but one and the same science can be formalized in different languages with different universes of discourse. The concept of the domain of a science and the concept of the universe of discourse of a formalization of a science are distinct, although they often coincide in extension. In order to analyse the presuppositions and implications of choices of domain and universe, this article discusses the treatment of omega arguments in three very different (...)
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  47.  13
    Upward Categoricity From a Successor Cardinal for Tame Abstract Classes with Amalgamation.Olivier Lessmann - 2005 - Journal of Symbolic Logic 70 (2):639 - 660.
    This paper is devoted to the proof of the following upward categoricity theorem: Let K be a tame abstract elementary class with amalgamation, arbitrarily large models, and countable Löwenheim-Skolem number. If K is categorical in ‮א‬₁ then K is categorical in every uncountable cardinal. More generally, we prove that if K is categorical in a successor cardinal λ⁺ then K is categorical everywhere above λ⁺.
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  48.  11
    Ultrafilters on $Omega$.James E. Baumgartner - 1995 - Journal of Symbolic Logic 60 (2):624-639.
    We study the $I$-ultrafilters on $\omega$, where $I$ is a collection of subsets of a set $X$, usually $\mathbb{R}$ or $\omega_1$. The $I$-ultrafilters usually contain the $P$-points, often as a small proper subset. We study relations between $I$-ultrafilters for various $I$, and closure of $I$-ultrafilters under ultrafilter sums. We consider, but do not settle, the question whether $I$-ultrafilters always exist.
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  49.  12
    Effective Categoricity of Abelian P -Groups.Wesley Calvert, Douglas Cenzer, Valentina S. Harizanov & Andrei Morozov - 2009 - Annals of Pure and Applied Logic 159 (1-2):187-197.
    We investigate effective categoricity of computable Abelian p-groups . We prove that all computably categorical Abelian p-groups are relatively computably categorical, that is, have computably enumerable Scott families of existential formulas. We investigate which computable Abelian p-groups are categorical and relatively categorical.
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  50.  10
    A Characterization of the $\Sigma_1$ -Definable Functions of $KP\Omega + $.Wolfgang Burr & Volker Hartung - 1998 - Archive for Mathematical Logic 37 (3):199-214.
    The subject of this paper is a characterization of the $\Sigma_1$ -definable set functions of Kripke-Platek set theory with infinity and a uniform version of axiom of choice: $KP\omega+(uniform\;AC)$ . This class of functions is shown to coincide with the collection of set functionals of type 1 primitive recursive in a given choice functional and $x\mapsto\omega$ . This goal is achieved by a Gödel Dialectica-style functional interpretation of $KP\omega+(uniform\;AC)$ and a computability proof for the involved functionals.
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