Results for 'Ultraproduct'

34 found
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  1.  11
    The Fundamental Theorem of Ultraproduct in Pavelka's Logic.Mingsheng Ying - 1992 - Mathematical Logic Quarterly 38 (1):197-201.
    In [This Zeitschrift 25 , 45-52, 119-134, 447-464], Pavelka systematically discussed propositional calculi with values in enriched residuated lattices and developed a general framework for approximate reasoning. In the first part of this paper we introduce the concept of generalized quantifiers into Pavelka's logic and establish the fundamental theorem of ultraproduct in first order Pavelka's logic with generalized quantifiers. In the second part of this paper we show that the fundamental theorem of ultraproduct in first order Pavelka's logic (...)
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  2.  13
    Saharon Shelah. On the Cardinality of Ultraproduct of Finite Sets. The Journal of Symbolic Logic, Vol. 35 , Pp. 83–84.A. Slomson - 1973 - Journal of Symbolic Logic 38 (4):650.
  3.  40
    On the Cardinality of Ultraproduct of Finite Sets.Saharon Shelah - 1970 - Journal of Symbolic Logic 35 (1):83-84.
  4.  31
    The Fundamental Theorem of Ultraproduct in Pavelka's Logic.Mingsheng Ying - 1992 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 38 (1):197-201.
  5.  25
    A Łoś Type Theorem for Linear Metric Formulas.Seyed-Mohammad Bagheri - 2010 - Mathematical Logic Quarterly 56 (1):78-84.
    We define an ultraproduct of metric structures based on a maximal probability charge and prove a variant of Łoś theorem for linear metric formulas. We also consider iterated ultraproducts.
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  6.  18
    On Topological Properties of Ultraproducts of Finite Sets.Gábor Sági & Saharon Shelah - 2005 - Mathematical Logic Quarterly 51 (3):254-257.
    In [3] a certain family of topological spaces was introduced on ultraproducts. These spaces have been called ultratopologies and their definition was motivated by model theory of higher order logics. Ultratopologies provide a natural extra topological structure for ultraproducts. Using this extra structure in [3] some preservation and characterization theorems were obtained for higher order logics. The purely topological properties of ultratopologies seem interesting on their own right. We started to study these properties in [2], where some questions remained open. (...)
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  7.  55
    Impossibility Results for Infinite-Electorate Abstract Aggregation Rules.Frederik Herzberg & Daniel Eckert - 2012 - Journal of Philosophical Logic 41 (1):273-286.
    Following Lauwers and Van Liedekerke (1995), this paper explores in a model-theoretic framework the relation between Arrovian aggregation rules and ultraproducts, in order to investigate a source of impossibility results for the case of an infinite number of individuals and an aggregation rule based on a free ultrafilter of decisive coalitions.
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  8.  25
    The Logic of Integration.Seyed-Mohammad Bagheri & Massoud Pourmahdian - 2009 - Archive for Mathematical Logic 48 (5):465-492.
    We develop a model theoretic framework for studying algebraic structures equipped with a measure. The real line is used as a value space and its usual arithmetical operations as connectives. Integration is used as a quantifier. We extend some basic results of pure model theory to this context and characterize measurable sets in terms of zero-sets of formulas.
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  9.  26
    A Hierarchy of Maps Between Compacta.Paul Bankston - 1999 - Journal of Symbolic Logic 64 (4):1628-1644.
    Let CH be the class of compacta (i.e., compact Hausdorff spaces), with BS the subclass of Boolean spaces. For each ordinal α and pair $\langle K,L\rangle$ of subclasses of CH, we define Lev ≥α K,L), the class of maps of level at least α from spaces in K to spaces in L, in such a way that, for finite α, Lev ≥α (BS,BS) consists of the Stone duals of Boolean lattice embeddings that preserve all prenex first-order formulas of quantifier rank (...)
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  10.  17
    Ultratopologies.Gábor Sági & János Gerlits - 2004 - Mathematical Logic Quarterly 50 (6):603-612.
    The notion of ultratopologies was introduced in [6] motivated by the model theory of first and higher order logics. In [6] we established some model theoretical applications of ultratopologies, for example, we provided a purely set theoretical characterization for classes de.nable by second order existential formulas. The present note deals with topological properties of ultratopologies, like density and compactness.
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  11.  28
    On Ultracoproducts of Compact Hausdorff Spaces.R. Gurevič - 1988 - Journal of Symbolic Logic 53 (1):294-300.
    I present solutions to several questions of Paul Bankston [2] by means of another version of the ultracoproduct construction, and explain the relation of ultracoproduct of compact Hausdorff spaces to other constructions combining topology, algebra and logic.
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  12.  27
    Standard Sets in Nonstandard Set Theory.Petr Andreev & Karel Hrbacek - 2004 - Journal of Symbolic Logic 69 (1):165-182.
    We prove that Standardization fails in every nontrivial universe definable in the nonstandard set theory BST, and that a natural characterization of the standard universe is both consistent with and independent of BST. As a consequence we obtain a formulation of nonstandard class theory in the ∈-language.
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  13.  16
    Ultraproducts and Higher Order Formulas.Gábor Sági - 2002 - Mathematical Logic Quarterly 48 (2):261-275.
    Which ultraproducts preserve the validity of formulas of higher order logics? To answer this question, we will introduce natural topologies on ultraproducts. We will show, that ultraproducts preserving certain higher order formulas can be characterized in terms of these topologies. As an application of the above results, we provide a constructive, purely model theoretic characterization for classes definable by second order existential formulas.
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  14.  15
    Regular Ultrapowers at Regular Cardinals.Juliette Kennedy, Saharon Shelah & Jouko Väänänen - 2015 - Notre Dame Journal of Formal Logic 56 (3):417-428.
    In earlier work by the first and second authors, the equivalence of a finite square principle $\square^{\mathrm{fin}}_{\lambda,D}$ with various model-theoretic properties of structures of size $\lambda $ and regular ultrafilters was established. In this paper we investigate the principle $\square^{\mathrm{fin}}_{\lambda,D}$—and thereby the above model-theoretic properties—at a regular cardinal. By Chang’s two-cardinal theorem, $\square^{\mathrm{fin}}_{\lambda,D}$ holds at regular cardinals for all regular filters $D$ if we assume the generalized continuum hypothesis. In this paper we prove in ZFC that, for certain regular filters (...)
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  15.  3
    Special ultrafilters and cofinal subsets of $$({}^omega omega, <^*)$$.Peter Nyikos - forthcoming - Archive for Mathematical Logic:1-18.
    The interplay between ultrafilters and unbounded subsets of \ with the order \ of strict eventual domination is studied. Among the tools are special kinds of non-principal ultrafilters on \. These include simple P-points; that is, ultrafilters with a base that is well-ordered with respect to the reverse of the order \ of almost inclusion. It is shown that the cofinality of such a base must be either \, the least cardinality of \-unbounded set, or \, the least cardinality of (...)
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  16.  11
    Theories Without the Tree Property of the Second Kind.Artem Chernikov - 2014 - Annals of Pure and Applied Logic 165 (2):695-723.
    We initiate a systematic study of the class of theories without the tree property of the second kind — NTP2. Most importantly, we show: the burden is “sub-multiplicative” in arbitrary theories ; NTP2 is equivalent to the generalized Kimʼs lemma and to the boundedness of ist-weight; the dp-rank of a type in an arbitrary theory is witnessed by mutually indiscernible sequences of realizations of the type, after adding some parameters — so the dp-rank of a 1-type in any theory is (...)
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  17.  23
    Pseudofinite Structures and Simplicity.Darío García, Dugald Macpherson & Charles Steinhorn - 2015 - Journal of Mathematical Logic 15 (1):1550002.
    We explore a notion of pseudofinite dimension, introduced by Hrushovski and Wagner, on an infinite ultraproduct of finite structures. Certain conditions on pseudofinite dimension are identified that guarantee simplicity or supersimplicity of the underlying theory, and that a drop in pseudofinite dimension is equivalent to forking. Under a suitable assumption, a measure-theoretic condition is shown to be equivalent to local stability. Many examples are explored, including vector spaces over finite fields viewed as 2-sorted finite structures, and homocyclic groups. Connections (...)
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  18.  16
    Dunn–Priest Quotients of Many-Valued Structures.Thomas Macaulay Ferguson - 2017 - Notre Dame Journal of Formal Logic 58 (2):221-239.
    J. Michael Dunn’s Theorem in 3-Valued Model Theory and Graham Priest’s Collapsing Lemma provide the means of constructing first-order, three-valued structures from classical models while preserving some control over the theories of the ensuing models. The present article introduces a general construction that we call a Dunn–Priest quotient, providing a more general means of constructing models for arbitrary many-valued, first-order logical systems from models of any second system. This technique not only counts Dunn’s and Priest’s techniques as special cases, but (...)
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  19.  68
    An Institution-Independent Proof of the Beth Definability Theorem.M. Aiguier & F. Barbier - 2007 - Studia Logica 85 (3):333-359.
    A few results generalizing well-known classical model theory ones have been obtained in institution theory these last two decades (e.g. Craig interpolation, ultraproduct, elementary diagrams). In this paper, we propose a generalized institution-independent version of the Beth definability theorem.
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  20.  48
    Strongly Representable Atom Structures of Cylindric Algebras.Robin Hirsch & Ian Hodkinson - 2009 - Journal of Symbolic Logic 74 (3):811-828.
    A cylindric algebra atom structure is said to be strongly representable if all atomic cylindric algebras with that atom structure are representable. This is equivalent to saying that the full complex algebra of the atom structure is a representable cylindric algebra. We show that for any finite n >3, the class of all strongly representable n-dimensional cylindric algebra atom structures is not closed under ultraproducts and is therefore not elementary. Our proof is based on the following construction. From an arbitrary (...)
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  21.  7
    Ordered Asymptotic Classes of Finite Structures.Darío García - 2020 - Annals of Pure and Applied Logic 171 (4):102776.
    We introduce the concept of o-asymptotic classes of finite structures, melding ideas coming from 1-dimensional asymptotic classes and o-minimality. Along with several examples and non-examples of these classes, we present some classification theory results of their infinite ultraproducts: Every infinite ultraproduct of structures in an o-asymptotic class is superrosy of U^þ-rank 1, and NTP2 (in fact, inp-minimal).
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  22.  7
    Countable Ultraproducts Without CH.Michael Canjar - 1988 - Annals of Pure and Applied Logic 37 (1):1-79.
    An important application of ultrafilters is in the ultraproduct construction in model theory. In this paper we study ultraproducts of countable structures, whose universe we assume is ω , using ultrafilters on a countable index set, which we also assume to be ω . Many of the properties of the ultraproduct are in fact inherent properties of the ultrafilter. For example, if we take a sequence of countable linear orders without maximal element, then their ultraproduct will have (...)
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  23.  20
    Reduced Coproducts of Compact Hausdorff Spaces.Paul Bankston - 1987 - Journal of Symbolic Logic 52 (2):404-424.
    By analyzing how one obtains the Stone space of the reduced product of an indexed collection of Boolean algebras from the Stone spaces of those algebras, we derive a topological construction, the "reduced coproduct", which makes sense for indexed collections of arbitrary Tichonov spaces. When the filter in question is an ultrafilter, we show how the "ultracoproduct" can be obtained from the usual topological ultraproduct via a compactification process in the style of Wallman and Frink. We prove theorems dealing (...)
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  24.  13
    Logic, Partial Orders and Topology.Hugo Mariano & Francisco Miraglia - 2005 - Manuscrito 28 (2):449-546.
    We give a version of L´os’ ultraproduct result for forcing in Kripke structures in a first-order language with equality and discuss ultrafilters in a topology naturally associated to a partial order. The presentation also includes background material so as to make the exposition accessible to those whose main interest is Computer Science, Artificial Intelligence and/or Philosophy.
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  25.  42
    A Note on Direct Products and Ultraproducts of Logical Matrices.Jan Zygmunt - 1974 - Studia Logica 33 (4):349 - 357.
    In this contribution we shall characterize matrix consequence operation determined by a direct product and an ultraproduct of a family of logical matrices. As an application we shall describe finite consequence operations with the help of ultrapowers.
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  26.  60
    External Automorphisms of Ultraproducts of Finite Models.Philipp Lücke & Saharon Shelah - 2012 - Archive for Mathematical Logic 51 (3-4):433-441.
    Let ${\fancyscript{L}}$ be a finite first-order language and ${\langle{\fancyscript{M}_n} \,|\, {n < \omega}\rangle}$ be a sequence of finite ${\fancyscript{L}}$ -models containing models of arbitrarily large finite cardinality. If the intersection of less than continuum-many dense open subsets of Cantor Space ω 2 is non-empty, then there is a non-principal ultrafilter ${\fancyscript{U}}$ over ω such that the corresponding ultraproduct ${\prod_\fancyscript{U}\fancyscript{M}_n}$ has an automorphism that is not induced by an element of ${\prod_{n<\omega}{\rm Aut}(\fancyscript{M}_n)}$.
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  27.  26
    On Models with Variable Universe.Bernd Ingo Dahn - 1975 - Studia Logica 34 (1):11 - 23.
    In this paper some parts of the model theory for logics based on generalised Kripke semantics are developed. Löwenheim-Skolem theorems and some applications of ultraproduct constructions for generalised Kripke models with variable universe are investigated using similar theorems of the model theory for classical logic. The results are generalizations of the theorems of [4].
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  28.  4
    Henselian Valued Fields and Inp-Minimality.Artem Chernikov & Pierre Simon - 2019 - Journal of Symbolic Logic 84 (4):1510-1526.
    We prove that every ultraproduct of p-adics is inp-minimal. More generally, we prove an Ax-Kochen type result on preservation of inp-minimality for Henselian valued fields of equicharacteristic 0 in the RV language.
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  29. Ultraproducts of SCI.Stephen Bloom & Roman Suszko - 1975 - Bulletin of the Section of Logic 4 (1):9-14.
    This note concerns the ultraproduct construction. It is observed that in the class of SCI models ultraproducts may be constructed by a method apparently dierent from the standard one. Since the standard models of veryday model theory form a subclass of SCI models, one obtains a new view of ultraproducts. Aside from a few remarks, the paper is self-contained.
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  30.  15
    On a Problem in Algebraic Model Theory.Bui Huy Hien - 1982 - Bulletin of the Section of Logic 11 (3/4):103-107.
    In Andreka-Nemeti [1] the class ST r of all small trees over C is dened for an arbitrary category C. Throughout the present paper C de- notes an arbitrary category. In Def. 4 of [1] on p. 367 the injectivity relation j= ) is dened. Intuitively the members of ST r represent the formulas and j= represents the validity relation be- tween objects of C considered as models and small trees of C considered as formulas. If ' 2 ST r (...)
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  31.  11
    Quantified Universes and Ultraproducts.Alireza Mofidi & Seyed-Mohammad Bagheri - 2012 - Mathematical Logic Quarterly 58 (1-2):63-74.
    A quantified universe is a set M equipped with a Riesz space equation image of real functions on Mn, for each n, and a second order operation equation image. Metric structures 4, graded probability structures 9 and many other structures in analysis are examples of such universes. We define ultraproduct of quantified universes and study properties preserved by this construction. We then discuss logics defined on the basis of classes of quantified universes which are closed under this construction.
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  32.  20
    Co-Critical Points of Elementary Embeddings.Michael Sheard - 1985 - Journal of Symbolic Logic 50 (1):220-226.
    Probably the two most famous examples of elementary embeddings between inner models of set theory are the embeddings of the universe into an inner model given by a measurable cardinal and the embeddings of the constructible universeLinto itself given by 0#. In both of these examples, the “target model” is a subclass of the “ground model”. It is not hard to find examples of embeddings in which the target model is not a subclass of the ground model: ifis a generic (...)
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  33.  11
    Observational Ultraproducts of Polynomial Coalgebras.Robert Goldblatt - 2003 - Annals of Pure and Applied Logic 123 (1-3):235-290.
    Coalgebras of polynomial functors constructed from sets of observable elements have been found useful in modelling various kinds of data types and state-transition systems. This paper continues the study of equational logic and model theory for polynomial coalgebras begun in Goldblatt , where it was shown that Boolean combinations of equations between terms of observable type form a natural language of observable formulas for specifying properties of polynomial coalgebras, and for giving a Hennessy–Milner style logical characterisation of observational indistinguishability of (...)
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  34.  5
    On Finite Approximations of Topological Algebraic Systems.L. Yu Glebsky, E. I. Gordon & C. Ward Hensen - 2007 - Journal of Symbolic Logic 72 (1):1 - 25.
    We introduce and discuss a concept of approximation of a topological algebraic system A by finite algebraic systems from a given class K. If A is discrete, this concept agrees with the familiar notion of a local embedding of A in a class K of algebraic systems. One characterization of this concept states that A is locally embedded in K iff it is a subsystem of an ultraproduct of systems from K. In this paper we obtain a similar characterization (...)
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