75 found
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  1.  22
    Quantum Team Logic and Bell’s Inequalities.Tapani Hyttinen, Gianluca Paolini & Jouko Väänänen - 2015 - Review of Symbolic Logic 8 (4):722-742.
    A logical approach to Bell's Inequalities of quantum mechanics has been introduced by Abramsky and Hardy [2]. We point out that the logical Bell's Inequalities of [2] are provable in the probability logic of Fagin, Halpern and Megiddo [4]. Since it is now considered empirically established that quantum mechanics violates Bell's Inequalities, we introduce a modified probability logic, that we call quantum team logic, in which Bell's Inequalities are not provable, and prove a Completeness Theorem for this logic. For this (...)
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  2.  9
    Independence in Finitary Abstract Elementary Classes.Tapani Hyttinen & Meeri Kesälä - 2006 - Annals of Pure and Applied Logic 143 (1-3):103-138.
    In this paper we study a specific subclass of abstract elementary classes. We construct a notion of independence for these AEC’s and show that under simplicity the notion has all the usual properties of first order non-forking over complete types. Our approach generalizes the context of 0-stable homogeneous classes and excellent classes. Our set of assumptions follow from disjoint amalgamation, existence of a prime model over 0/, Löwenheim–Skolem number being ω, -tameness and a property we call finite character. We also (...)
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  3.  18
    Beyond Abstract Elementary Classes: On the Model Theory of Geometric Lattices.Tapani Hyttinen & Gianluca Paolini - 2018 - Annals of Pure and Applied Logic 169 (2):117-145.
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  4.  24
    Interpreting Groups and Fields in Some Nonelementary Classes.Tapani Hyttinen, Olivier Lessmann & Saharon Shelah - 2005 - Journal of Mathematical Logic 5 (1):1-47.
    This paper is concerned with extensions of geometric stability theory to some nonelementary classes. We prove the following theorem: Theorem. Let [Formula: see text] be a large homogeneous model of a stable diagram D. Let p, q ∈ SD, where p is quasiminimal and q unbounded. Let [Formula: see text] and [Formula: see text]. Suppose that there exists an integer n < ω such that [Formula: see text] for any independent a1, …, an ∈ P and finite subset C ⊆ (...)
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  5.  10
    Strong Splitting in Stable Homogeneous Models.Tapani Hyttinen & Saharon Shelah - 2000 - Annals of Pure and Applied Logic 103 (1-3):201-228.
    In this paper we study elementary submodels of a stable homogeneous structure. We improve the independence relation defined in Hyttinen 167–182). We apply this to prove a structure theorem. We also show that dop and sdop are essentially equivalent, where the negation of dop is the property we use in our structure theorem and sdop implies nonstructure, see Hyttinen.
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  6.  76
    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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  7.  28
    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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  8.  8
    Quasiminimal Structures, Groups and Zariski-Like Geometries.Tapani Hyttinen & Kaisa Kangas - 2016 - Annals of Pure and Applied Logic 167 (6):457-505.
  9.  8
    Constructing Strongly Equivalent Nonisomorphic Models for Unstable Theories.Tapani Hyttinen & Heikki Tuuri - 1991 - Annals of Pure and Applied Logic 52 (3):203-248.
    If T is an unstable theory of cardinality <λ or countable stable theory with OTOP or countable superstable theory with DOP, λω λω1 in the superstable with DOP case) is regular and λ<λ=λ, then we construct for T strongly equivalent nonisomorphic models of cardinality λ. This can be viewed as a strong nonstructure theorem for such theories. We also consider the case when T is unsuperstable and develop further a result of Shelah about the existence of L∞,λ-equivalent nonisomorphic models for (...)
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  10.  14
    On Σ11-Complete Equivalence Relations on the Generalized Baire Space.Tapani Hyttinen & Vadim Kulikov - 2015 - Mathematical Logic Quarterly 61 (1-2):66-81.
  11.  16
    A Rank for the Class of Elementary Submodels of a Superstable Homogeneous Model.Tapani Hyttinen & Olivier Lessmann - 2002 - Journal of Symbolic Logic 67 (4):1469-1482.
    We study the class of elementary submodels of a large superstable homogeneous model. We introduce a rank which is bounded in the superstable case, and use it to define a dependence relation which shares many (but not all) of the properties of forking in the first order case. The main difference is that we do not have extension over all sets. We also present an example of Shelah showing that extension over all sets may not hold for any dependence relation (...)
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  12.  8
    Generalizing Morley's Theorem.Tapani Hyttinen - 1998 - Mathematical Logic Quarterly 44 (2):176-184.
    We study the categoricity of the classes of elementary submodels of a homogeneous structure.
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  13.  66
    Henkin Quantifiers and the Definability of Truth.Tapani Hyttinen & Gabriel Sandu - 2000 - Journal of Philosophical Logic 29 (5):507-527.
    Henkin quantifiers have been introduced in Henkin (1961). Walkoe (1970) studied basic model-theoretical properties of an extension $L_{*}^{1}$ (H) of ordinary first-order languages in which every sentence is a first-order sentence prefixed with a Henkin quantifier. In this paper we consider a generalization of Walkoe's languages: we close $L_{*}^{1}$ (H) with respect to Boolean operations, and obtain the language L¹(H). At the next level, we consider an extension $L_{*}^{2}$ (H) of L¹(H) in which every sentence is an L¹(H)-sentence prefixed with (...)
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  14.  13
    On the $\Kappa$ -Cub Game on $\Lambda $ and $I[\Lambda ]$.Taneli Huuskonen, Tapani Hyttinen & Mika Rautila - 1999 - Archive for Mathematical Logic 38 (8):549-557.
    We discuss the relationships between the notions of $\kappa $ -cub game on $\lambda $ , $\kappa $ -cub subset of $\lambda $ , the ideal of good subsets of $\lambda $ and the problem of adding a $\kappa $ -cub into a given $\kappa $ -stationary subset of $\lambda $ . We also give a short introduction to the ideal of good subsets of $\lambda $.
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  15.  7
    Categoricity and Universal Classes.Tapani Hyttinen & Kaisa Kangas - 2018 - Mathematical Logic Quarterly 64 (6):464-477.
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  16. Lascar Types and Lascar Automorphisms in Abstract Elementary Classes.Tapani Hyttinen & Meeri Kesälä - 2011 - Notre Dame Journal of Formal Logic 52 (1):39-54.
    We study Lascar strong types and Galois types and especially their relation to notions of type which have finite character. We define a notion of a strong type with finite character, the so-called Lascar type. We show that this notion is stronger than Galois type over countable sets in simple and superstable finitary AECs. Furthermore, we give an example where the Galois type itself does not have finite character in such a class.
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  17.  64
    IF Logic and the Foundations of Mathematics.Gabriel Sandu & Tapani Hyttinen - 2001 - Synthese 126 (1):37-47.
  18.  8
    Simplicity and Uncountable Categoricity in Excellent Classes.Tapani Hyttinen & Olivier Lessmann - 2006 - Annals of Pure and Applied Logic 139 (1):110-137.
    We introduce Lascar strong types in excellent classes and prove that they coincide with the orbits of the group generated by automorphisms fixing a model. We define a new independence relation using Lascar strong types and show that it is well-behaved over models, as well as over finite sets. We then develop simplicity and show that, under simplicity, the independence relation satisfies all the properties of nonforking in a stable first order theory. Further, simplicity for an excellent class, as well (...)
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  19.  10
    On Stability in Finite Models.Tapani Hyttinen - 2000 - Archive for Mathematical Logic 39 (2):89-102.
    We search for a set-up in which results from the theory of infinite models hold for finite models. As an example we prove results from stability theory.
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  20.  12
    A Logic for Arguing About Probabilities in Measure Teams.Tapani Hyttinen, Gianluca Paolini & Jouko Väänänen - 2017 - Archive for Mathematical Logic 56 (5-6):475-489.
    We use sets of assignments, a.k.a. teams, and measures on them to define probabilities of first-order formulas in given data. We then axiomatise first-order properties of such probabilities and prove a completeness theorem for our axiomatisation. We use the Hardy–Weinberg Principle of biology and the Bell’s Inequalities of quantum physics as examples.
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  21.  28
    On Scott and Karp Trees of Uncountable Models.Tapani Hyttinen & Jouko Väänänen - 1990 - Journal of Symbolic Logic 55 (3):897-908.
    Let U and B be two countable relational models of the same first order language. If the models are nonisomorphic, there is a unique countable ordinal α with the property that $\mathfrak{U} \equiv^\alpha_{\infty\omega} \mathfrak{B} \text{but not} \mathfrak{U} \equiv^{\alpha + 1}_{\infty\omega} \mathfrak{B},$ i.e. U and B are L ∞ω -equivalent up to quantifier-rank α but not up to α + 1. In this paper we consider models U and B of cardinality ω 1 and construct trees which have a similar relation (...)
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  22.  10
    A Generalized Borel-Reducibility Counterpart of Shelah’s Main Gap Theorem.Tapani Hyttinen, Vadim Kulikov & Miguel Moreno - 2017 - Archive for Mathematical Logic 56 (3-4):175-185.
    We study the κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document}-Borel-reducibility of isomorphism relations of complete first order theories in a countable language and show the consistency of the following: For all such theories T and T′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^{\prime }$$\end{document}, if T is classifiable and T′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^{\prime }$$\end{document} is not, then the isomorphism of models of T′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} (...)
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  23.  6
    On the Reducibility of Isomorphism Relations.Tapani Hyttinen & Miguel Moreno - 2017 - Mathematical Logic Quarterly 63 (3-4):175-192.
    We study the Borel reducibility of isomorphism relations in the generalized Baire space math formula. In the main result we show for inaccessible κ, that if T is a classifiable theory and math formula is stable with the orthogonal chain property, then the isomorphism of models of T is Borel reducible to the isomorphism of models of math formula.
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  24.  6
    Finiteness of U-Rank Implies Simplicity in Homogeneous Structures.Tapani Hyttinen - 2003 - Mathematical Logic Quarterly 49 (6):576.
    A superstable homogeneous structure is said to be simple if every complete type over any set A has a free extension over any B ⊇ A. In this paper we give a characterization for this property in terms of U-rank. As a corollary we get that if the structure has finite U-rank, then it is simple.
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  25.  18
    Canonical Finite Diagrams and Quantifier Elimination.Tapani Hyttinen - 2002 - Mathematical Logic Quarterly 48 (4):533-554.
    We revisit the theory of amalgamation classes but we do not insist on staying within elementary classes.
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  26.  10
    Finitely Generated Submodels of an Uncountably Categorical Homogeneous Structure.Tapani Hyttinen - 2004 - Mathematical Logic Quarterly 50 (1):77.
    We generalize the result of non-finite axiomatizability of totally categorical first-order theories from elementary model theory to homogeneous model theory. In particular, we lift the theory of envelopes to homogeneous model theory and develope theory of imaginaries in the case of ω-stable homogeneous classes of finite U-rank.
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  27.  8
    On Potential Isomorphism and Non-Structure.Taneli Huuskonen, Tapani Hyttinen & Mika Rautila - 2004 - Archive for Mathematical Logic 43 (1):85-120.
    We show in the paper that for any non-classifiable countable theory T there are non-isomorphic models and that can be forced to be isomorphic without adding subsets of small cardinality. By making suitable cardinal arithmetic assumptions we can often preserve stationary sets as well. We also study non-structure theorems relative to the Ehrenfeucht-Fraïssé game.
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  28.  13
    On the Number of Elementary Submodels of an Unsuperstable Homogeneous Structure.Tapani Hyttinen & Saharon Shelah - 1998 - Mathematical Logic Quarterly 44 (3):354-358.
    We show that if M is a stable unsuperstable homogeneous structure, then for most κ ⩽ |M|, the number of elementary submodels of M of power κ is 2κ.
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  29.  3
    Reduction of Database Independence to Dividing in Atomless Boolean Algebras.Tapani Hyttinen & Gianluca Paolini - 2016 - Archive for Mathematical Logic 55 (3-4):505-518.
    We prove that the form of conditional independence at play in database theory and independence logic is reducible to the first-order dividing calculus in the theory of atomless Boolean algebras. This establishes interesting connections between independence in database theory and stochastic independence. As indeed, in light of the aforementioned reduction and recent work of Ben-Yaacov :957–1012, 2013), the former case of independence can be seen as the discrete version of the latter.
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  30.  45
    On Borel Equivalence Relations in Generalized Baire Space.Sy-David Friedman & Tapani Hyttinen - 2012 - Archive for Mathematical Logic 51 (3-4):299-304.
    We construct two Borel equivalence relations on the generalized Baire space κ κ , κ <κ = κ > ω, with the property that neither of them is Borel reducible to the other. A small modification of the construction shows that the straightforward generalization of the Glimm-Effros dichotomy fails.
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  31.  57
    Deflationism and Arithmetical Truth.Tapani Hyttinen & Gabriel Sandu - 2004 - Dialectica 58 (3):413–426.
  32.  15
    The Nonabsoluteness of Model Existence in Uncountable Cardinals for $L{Omega{1},Omega}$.Sy-David Friedman, Tapani Hyttinen & Martin Koerwien - 2013 - Notre Dame Journal of Formal Logic 54 (2):137-151.
    For sentences $\phi$ of $L_{\omega_{1},\omega}$, we investigate the question of absoluteness of $\phi$ having models in uncountable cardinalities. We first observe that having a model in $\aleph_{1}$ is an absolute property, but having a model in $\aleph_{2}$ is not as it may depend on the validity of the continuum hypothesis. We then consider the generalized continuum hypothesis context and provide sentences for any $\alpha\in\omega_{1}\setminus\{0,1,\omega\}$ for which the existence of a model in $\aleph_{\alpha}$ is nonabsolute . Finally, we present a complete (...)
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  33.  6
    On the [Mathematical Formula]-Cub Game on [Mathematical Formula] and [Mathematical Formula].Taneli Huuskonen, Tapani Hyttinen & Mika Rautila - 1999 - Archive for Mathematical Logic 38 (8):549-557.
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  34.  4
    Remarks on Structure Theorems for $\Omega_{1}$ -Saturated Models.Tapani Hyttinen - 1995 - Notre Dame Journal of Formal Logic 36 (2):269-278.
    We give a characterization for those stable theories whose $\omega_{1}$-saturated models have a "Shelah-style" structure theorem. We use this characterization to prove that if a theory is countable, stable, and 1-based without dop or didip, then its $\omega_{1}$-saturated models have a structure theorem. Prior to us, this is proved in a paper of Hart, Pillay, and Starchenko . Some other remarks are also included.
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  35. Constructing Strongly Equivalent Nonisomorphic Models for Unsuperstable Theories, Part A.Tapani Hyttinen & Saharon Shelah - 1994 - Journal of Symbolic Logic 59 (3):984-996.
    We study how equivalent nonisomorphic models an unsuperstable theory can have. We measure the equivalence by Ehrenfeucht-Fraisse games. This paper continues the work started in $[HT]$.
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  36.  9
    Types in Abstract Elementary Classes.Tapani Hyttinen - 2004 - Notre Dame Journal of Formal Logic 45 (2):99-108.
    We suggest a method of finding a notion of type to abstract elementary classes and determine under what assumption on these types the class has a well-behaved homogeneous and universal "monster" model, where homogeneous and universal are defined relative to our notion of type.
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  37.  25
    The Canary Tree Revisited.Tapani Hyttinen & Mika Rautila - 2001 - Journal of Symbolic Logic 66 (4):1677-1694.
    We generalize the result of Mekler and Shelah [3] that the existence of a canary tree is independent of ZFC + GCH to uncountable regular cardinals. We also correct an error from the original proof.
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  38.  12
    Onκ-Complete Reduced Products.Tapani Hyttinen - 1992 - Archive for Mathematical Logic 31 (3):193-199.
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  39.  6
    On Nonstructure of Elementary Submodels of an Unsuperstable Homogeneous Structure.Tapani Hyttinen - 1997 - Mathematical Logic Quarterly 43 (1):134-142.
    In the first part of this paper we let M be a stable homogeneous model and we prove a nonstructure theorem for the class of all elementary submodels of M, assuming that M is ‘unsuperstable’ and has Skolem functions. In the second part we assume that M is an unstable homogeneous model of large cardinality and we prove a nonstructure theorem for the class of all elementary submodels of M.
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  40.  3
    Interpreting Groups and Fields in Simple, Finitary AECs.Tapani Hyttinen & Meeri Kesälä - 2012 - Annals of Pure and Applied Logic 163 (9):1141-1162.
  41.  24
    On Non-Determined Ehrenfeucht-Fraïssé Games and Unstable Theories.Tapani Hyttinen & T. Hyttinen - 1992 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 38 (1):399-408.
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  42.  7
    Canonical Bases in Excellent Classes.Tapani Hyttinen & Olivier Lessmann - 2008 - Journal of Symbolic Logic 73 (1):165-180.
    We show that any (atomic) excellent class K can be expanded with hyperimaginaries to form an (atomic) excellent class Keq which has canonical bases. When K is, in addition, of finite U-rank, then Keq is also simple and has a full canonical bases theorem. This positive situation contrasts starkly with homogeneous model theory for example, where the eq-expansion may fail to be homogeneous. However, this paper shows that expanding an ω-stable, homogeneous class K gives rise to an excellent class, which (...)
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  43.  22
    Main Gap for Locally Saturated Elementary Submodels of a Homogeneous Structure.Tapani Hyttinen & Saharon Shelah - 2001 - Journal of Symbolic Logic 66 (3):1286-1302.
    We prove a main gap theorem for locally saturated submodels of a homogeneous structure. We also study the number of locally saturated models, which are not elementarily embeddable into each other.
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  44.  4
    Locally Modular Geometries in Homogeneous Structures.Tapani Hyttinen - 2005 - Mathematical Logic Quarterly 51 (3):291.
    We show that if M is a strongly minimal large homogeneous structure in a countable similarity type and the pregeometry of M is locally modular but not modular, then the pregeometry is affine over a division ring.
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  45.  8
    A Remark on Weakly Compact Cardinals.Tapani Hyttinen - 2002 - Mathematical Logic Quarterly 48 (3):397-402.
    We show that if κ is weakly compact, then κ → 3holds for treelike partitions. As an application we study model constructions.
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  46.  5
    Constructing Strongly Equivalent Nonisomorphic Models for Unsuperstable Theories. Part B.Tapani Hyttinen & Saharon Shelah - 1995 - Journal of Symbolic Logic 60 (4):1260-1272.
    We study how equivalent nonisomorphic models of unsuperstable theories can be. We measure the equivalence by Ehrenfeucht-Fraisse games. This paper continues [HS].
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  47.  7
    Remarks on Strong Nonstructure Theorems.Tapani Hyttinen, Saharon Shelah & Heikki Tuuri - 1993 - Notre Dame Journal of Formal Logic 34 (2):157-168.
  48.  4
    Remarks on Structure Theorems for -Saturated Models.Tapani Hyttinen - 1995 - Notre Dame Journal of Formal Logic 36 (2):269-278.
    We give a characterization for those stable theories whose -saturated models have a "Shelah-style" structure theorem. We use this characterization to prove that if a theory is countable, stable, and 1-based without dop or didip, then its -saturated models have a structure theorem. Prior to us, this is proved in a paper of Hart, Pillay, and Starchenko (in which they also count the number of models, which we do not do here). Some other remarks are also included.
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  49.  7
    Reducibility of Equivalence Relations Arising From Nonstationary Ideals Under Large Cardinal Assumptions.David Asperó, Tapani Hyttinen, Vadim Kulikov & Miguel Moreno - 2019 - Notre Dame Journal of Formal Logic 60 (4):665-682.
    Working under large cardinal assumptions such as supercompactness, we study the Borel reducibility between equivalence relations modulo restrictions of the nonstationary ideal on some fixed cardinal κ. We show the consistency of Eλ-clubλ++,λ++, the relation of equivalence modulo the nonstationary ideal restricted to Sλλ++ in the space λ++, being continuously reducible to Eλ+-club2,λ++, the relation of equivalence modulo the nonstationary ideal restricted to Sλ+λ++ in the space 2λ++. Then we show that for κ ineffable Ereg2,κ, the relation of equivalence modulo (...)
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  50.  13
    Classification Theory and 0#.Sy D. Friedman, Tapani Hyttinen & Mika Rautila - 2003 - Journal of Symbolic Logic 68 (2):580-588.
    We characterize the classifiability of a countable first-order theory T in terms of the solvability of the potential-isomorphism problem for models of T.
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