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  1. Vera Koponen & Tapani Hyttinen (2015). On Compactness of Logics That Can Express Properties of Symmetry or Connectivity. Studia Logica 103 (1):1-20.
    A condition, in two variants, is given such that if a property P satisfies this condition, then every logic which is at least as strong as first-order logic and can express P fails to have the compactness property. The result is used to prove that for a number of natural properties P speaking about automorphism groups or connectivity, every logic which is at least as strong as first-order logic and can express P fails to have the compactness property. The basic (...)
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  2. Sy-David Friedman, Tapani Hyttinen & Martin Koerwien (2013). The Nonabsoluteness of Model Existence in Uncountable Cardinals for $L{Omega{1},Omega}$. 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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  3. Sy-David Friedman & Tapani Hyttinen (2012). On Borel Equivalence Relations in Generalized Baire Space. 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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  4. Tapani Hyttinen & Meeri Kesälä (2012). Interpreting Groups and Fields in Simple, Finitary AECs. Annals of Pure and Applied Logic 163 (9):1141-1162.
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  5. Sy-David Friedman, Tapani Hyttinen & Agatha C. Walczak-Typke (2011). Potential Isomorphism of Elementary Substructures of a Strictly Stable Homogeneous Model. Journal of Symbolic Logic 76 (3):987 - 1004.
    The results herein form part of a larger project to characterize the classification properties of the class of submodels of a homogeneous stable diagram in terms of the solvability (in the sense of [1]) of the potential isomorphism problem for this class of submodels. We restrict ourselves to locally saturated submodels of the monster model m of some power π. We assume that in Gödel's constructible universe , π is a regular cardinal at least the successor of the first cardinal (...)
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  6. Tapani Hyttinen & Meeri Kesälä (2011). Categoricity Transfer in Simple Finitary Abstract Elementary Classes. 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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  7. Tapani Hyttinen & Meeri Kesälä (2010). Lascar Types and Lascar Automorphisms in Abstract Elementary Classes. 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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  8. Åsa Hirvonen & Tapani Hyttinen (2009). Categoricity in Homogeneous Complete Metric Spaces. 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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  9. Tapani Hyttinen & Olivier Lessmann (2008). Canonical Bases in Excellent Classes. 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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  10. Tapani Hyttinen (2006). Remark on Spectrums of Formulas with Henkin Quantifiers. Acta Philosophica Fennica 78:79.
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  11. Tapani Hyttinen (2006). Uncountably Categorical Local Tame Abstract Elementary Classes with Disjoint Amalgamation. Archive for Mathematical Logic 45 (1):63-73.
    We prove Baldwin-Lachlan theorem for local (LS(K)-)tame abstract elementary classes K with disjoint amalgamation property and with LS(K)=ω.
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  12. Tapani Hyttinen & Meeri Kesälä (2006). Independence in Finitary Abstract Elementary Classes. Annals of Pure and Applied Logic 143 (1):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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  13. Tapani Hyttinen & Olivier Lessmann (2006). Simplicity and Uncountable Categoricity in Excellent Classes. 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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  14. Tapani Hyttinen (2005). Locally Modular Geometries in Homogeneous Structures. 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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  15. Tapani Hyttinen, Olivier Lessmann & Saharon Shelah (2005). Interpreting Groups and Fields in Some Nonelementary Classes. Journal of Mathematical Logic 5 (01):1-47.
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  16. Taneli Huuskonen, Tapani Hyttinen & Mika Rautila (2004). On Potential Isomorphism and Non-Structure. 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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  17. Tapani Hyttinen (2004). Finitely Generated Submodels of an Uncountably Categorical Homogeneous Structure. 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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  18. Tapani Hyttinen (2004). Types in Abstract Elementary Classes. 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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  19. Tapani Hyttinen & Gabriel Sandu (2004). Deflationism and Arithmetical Truth. Dialectica 58 (3):413–426.
  20. Tapani Hyttinen & Gabriel Sandu (2004). Truth and Definite Truth. Annals of Pure and Applied Logic 126 (1-3):49-55.
    In this paper we consider truth as a vague predicate and inquire into the relation between truth and definite truth. We use some tools from modal logic to clarify this distinction, as done in McGee . Finally, we consider the question whether some of the results given by McGee can be transferred to the case in which the underlying logic is stronger than first-order logic. The result will be seen to be negative.
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  21. Sy D. Friedman, Tapani Hyttinen & Mika Rautila (2003). Classification Theory and $0^\#$. Journal of Symbolic Logic 68 (2): 580- 588.
    We characterize the classifiability of a countable first-order theory T in terms of the solvability (in the sense of [2]) of the potential-isomorphism problem for models of T.
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  22. Tapani Hyttinen (2003). Finiteness of U-Rank Implies Simplicity in Homogeneous Structures. 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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  23. Tapani Hyttinen (2003). Interpreting Groups Inside Modular Strongly Minimal Homogeneous Models. Journal of Mathematical Logic 3 (01):127-142.
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  24. Tapani Hyttinen (2002). A Remark on Weakly Compact Cardinals. 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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  25. Tapani Hyttinen (2002). Canonical Finite Diagrams and Quantifier Elimination. 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. Tapani Hyttinen & Olivier Lessmann (2002). A Rank for the Class of Elementary Submodels of a Superstable Homogeneous Model. 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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  27. Tapani Hyttinen & Mika Rautila (2001). The Canary Tree Revisited. 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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  28. Tapani Hyttinen & Saharon Shelah (2001). Main Gap for Locally Saturated Elementary Submodels of a Homogeneous Structure. 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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  29. Gabriel Sandu & Tapani Hyttinen (2001). IF Logic and the Foundations of Mathematics. Synthese 126 (1-2):37 - 47.
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  30. Tapani Hyttinen (2000). On Stability in Finite Models. 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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  31. Tapani Hyttinen & Gabriel Sandu (2000). Henkin Quantifiers and the Definability of Truth. 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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  32. Tapani Hyttinen & Saharon Shelah (2000). Strong Splitting in Stable Homogeneous Models. 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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  33. Taneli Huuskonen, Tapani Hyttinen & Mika Rautila (1999). On the Kappa-Cub Game on Lambda and I[Lambda ]. 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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  34. Taneli Huuskonen, Tapani Hyttinen & Mika Rautila (1999). On the [Mathematical Formula]-Cub Game on [Mathematical Formula] and [Mathematical Formula]. Archive for Mathematical Logic 8.
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  35. Tapani Hyttinen (1999). Stability and General Logics. Mathematical Logic Quarterly 45 (2):219-240.
    In this paper we make an attempt to study classes of models by using general logics. We do not believe that Lww is always the best logic for analyzing a class of models. Let K be a class of models and L a logic. The main assumptions we make about K and C are that K has the L-amalgamation property and, later in the paper, that K does not omit L-types. We show that, if modified suitably, most of the results (...)
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  36. Tapani Hyttinen & Saharon Shelah (1999). Constructing Strongly Equivalent Nonisomorphic Models for Unsuperstable Theories, Part C. Journal of Symbolic Logic 64 (2):634-642.
    In this paper we prove a strong nonstructure theorem for κ(T)-saturated models of a stable theory T with dop. This paper continues the work started in [1].
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  37. Tapani Hyttinen (1998). Generalizing Morley's Theorem. 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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  38. Tapani Hyttinen (1998). A Remark on Algebraic Closure and Orthogonality. Notre Dame Journal of Formal Logic 39 (4):527-530.
    We show that if is a stable theory with ndop and ndidip, then -primary models over free trees are -minimal over the tree. As a corollary we show, for example, that if is a stable theory and for all nonempty , , then is superstable or it has dop or didip.
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  39. Tapani Hyttinen & Saharon Shelah (1998). On the Number of Elementary Submodels of an Unsuperstable Homogeneous Structure. 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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  40. Tapani Hyttinen (1997). On Nonstructure of Elementary Submodels of an Unsuperstable Homogeneous Structure. 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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  41. Tapani Hyttinen (1996). Forking and Incomplete Types. Mathematical Logic Quarterly 42 (1):421-432.
    Let Δ be a set of formulas. In this paper we study the following question: under what assumptions on Δ, the concept “a complete Δ-type p over B does not fork over A ⊆ B” behaves well. We apply the results to the structure theory of ω1-saturated models.
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  42. Tapani Hyttinen (1995). Remarks on Structure Theorems for -Saturated Models. 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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  43. Tapani Hyttinen (1995). Remarks on Structure Theorems for $\Omega_{1}$ -Saturated Models. 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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  44. Tapani Hyttinen & Saharon Shelah (1995). Constructing Strongly Equivalent Nonisomorphic Models for Unsuperstable Theories. Part B. Journal of Symbolic Logic 60 (4):1260-1272.
    In this paper we prove a strong nonstructure theorem for κ(T)-saturated models of a stable theory T with dop. This paper continues the work started in [1].
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  45. Tapani Hyttinen & Saharon Shelah (1994). Constructing Strongly Equivalent Nonisomorphic Models for Unsuperstable Theories, Part A. Journal of Symbolic Logic 59 (3):984-996.
    In this paper we prove a strong nonstructure theorem for κ(T)-saturated models of a stable theory T with dop. This paper continues the work started in [1].
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  46. Tapani Hyttinen, Saharon Shelah & Heikki Tuuri (1993). Remarks on Strong Nonstructure Theorems. Notre Dame Journal of Formal Logic 34 (2):157-168.
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  47. Tapani Hyttinen (1992). Onκ-Complete Reduced Products. Archive for Mathematical Logic 31 (3):193-199.
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  48. Tapani Hyttinen & T. Hyttinen (1992). On Non‐Determined Ehrenfeucht‐Fraïssé Games and Unstable Theories. Mathematical Logic Quarterly 38 (1):399-408.
    In this paper we prove under some set theoretical assumptions that if T is a countable unstable theory then there is a pair of models of T such that Ehrenfeucht-Fraïssé games between these models of large variety of lengths are non-determined.
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  49. Tapani Hyttinen (1991). Preservation by Homomorphisms and Infinitary Languages. Notre Dame Journal of Formal Logic 32 (2):167-172.
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  50. Tapani Hyttinen & Heikki Tuuri (1991). Constructing Strongly Equivalent Nonisomorphic Models for Unstable Theories. 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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