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152Second-order logic and foundations of mathematicsBulletin of Symbolic Logic 7 (4): 504-520. 2001.We discuss the differences between first-order set theory and second-order logic as a foundation for mathematics. We analyse these languages in terms of two levels of formalization. The analysis shows that if second-order logic is understood in its full semantics capable of characterizing categorically central mathematical concepts, it relies entirely on informal reasoning. On the other hand, if it is given a weak semantics, it loses its power in expressing concepts categorically. First-order se…Read more
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76Barwise: Abstract model theory and generalized quantifiersBulletin of Symbolic Logic 10 (1): 37-53. 2004.§1. Introduction. After the pioneering work of Mostowski [29] and Lindström [23] it was Jon Barwise's papers [2] and [3] that brought abstract model theory and generalized quantifiers to the attention of logicians in the early seventies. These papers were greeted with enthusiasm at the prospect that model theory could be developed by introducing a multitude of extensions of first order logic, and by proving abstract results about relationships holding between properties of these logics. Examples…Read more
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244The hierarchy theorem for generalized quantifiersJournal of Symbolic Logic 61 (3): 802-817. 1996.The concept of a generalized quantifier of a given similarity type was defined in [12]. Our main result says that on finite structures different similarity types give rise to different classes of generalized quantifiers. More exactly, for every similarity type t there is a generalized quantifier of type t which is not definable in the extension of first order logic by all generalized quantifiers of type smaller than t. This was proved for unary similarity types by Per Lindström [17] with a count…Read more
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64Stationary sets and infinitary logicJournal of Symbolic Logic 65 (3): 1311-1320. 2000.Let K 0 λ be the class of structures $\langle\lambda, , where $A \subseteq \lambda$ is disjoint from a club, and let K 1 λ be the class of structures $\langle\lambda, , where $A \subseteq \lambda$ contains a club. We prove that if $\lambda = \lambda^{ is regular, then no sentence of L λ+κ separates K 0 λ and K 1 λ . On the other hand, we prove that if $\lambda = \mu^+,\mu = \mu^{ , and a forcing axiom holds (and ℵ L 1 = ℵ 1 if μ = ℵ 0 ), then there is a sentence of L λλ which separates K 0 λ and…Read more
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25Pursuing Logic without BordersIn Åsa Hirvonen, Juha Kontinen, Roman Kossak & Andrés Villaveces (eds.), Logic Without Borders: Essays on Set Theory, Model Theory, Philosophical Logic and Philosophy of Mathematics, De Gruyter. pp. 403-416. 2015.
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36On the symbiosis between model-theoretic and set-theoretic properties of large cardinalsJournal of Symbolic Logic 81 (2): 584-604. 2016.
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14Finite information logicAnnals of Pure and Applied Logic 134 (1): 83-93. 2005.We introduce a generalization of Independence Friendly logic in which Eloise is restricted to a finite amount of information about Abelard’s moves. This logic is shown to be equivalent to a sublogic of first-order logic, to have the finite model property, and to be decidable. Moreover, it gives an exponential compression relative to logic
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60Trees and Π 1 1 -Subsets of ω1 ω 1Journal of Symbolic Logic 58 (3). 1993.We study descriptive set theory in the space ω1 ω 1 by letting trees with no uncountable branches play a similar role as countable ordinals in traditional descriptive set theory. By using such trees, we get, for example, a covering property for the class of Π 1 1 -sets of ω1 ω 1 . We call a family U of trees universal for a class V of trees if $\mathscr{U} \subseteq \mathscr{V}$ and every tree in V can be order-preservingly mapped into a tree in U. It is well known that the class of countable tr…Read more
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20Regular Ultrapowers at Regular CardinalsNotre Dame Journal of Formal Logic 56 (3): 417-428. 2015.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 a…Read more
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15This book comprises revised full versions of lectures given during the 9th European Summer School in Logic, Languages, and Information, ESSLLI'97, held in Aix-en-Provence, France, in August 1997. The six lectures presented introduce the reader to the state of the art in the area of generalized quantifiers and computation. Besides an introductory survey by the volume editor various aspects of generalized quantifiers are studied in depth.
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70Unary quantifiers on finite modelsJournal of Logic, Language and Information 6 (3): 275-304. 1997.In this paper (except in Section 5) all quantifiers are assumedto be so called simple unaryquantifiers, and all models are assumedto be finite. We give a necessary and sufficientcondition for a quantifier to be definablein terms of monotone quantifiers. For amonotone quantifier we give a necessaryand sufficient condition for beingdefinable in terms of a given set of bounded monotonequantifiers. Finally, we give a necessaryand sufficient condition for a monotonequantifier to be definable in terms…Read more
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155Dependence and IndependenceStudia Logica 101 (2): 399-410. 2013.We introduce an atomic formula ${\vec{y} \bot_{\vec{x}}\vec{z}}$ intuitively saying that the variables ${\vec{y}}$ are independent from the variables ${\vec{z}}$ if the variables ${\vec{x}}$ are kept constant. We contrast this with dependence logic ${\mathcal{D}}$ based on the atomic formula = ${(\vec{x}, \vec{y})}$ , actually equivalent to ${\vec{y} \bot_{\vec{x}}\vec{y}}$ , saying that the variables ${\vec{y}}$ are totally determined by the variables ${\vec{x}}$ . We show that ${\vec{y} \bot_{…Read more
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148On the number of automorphisms of uncountable modelsJournal of Symbolic Logic 58 (4): 1402-1418. 1993.Let σ(U) denote the number of automorphisms of a model U of power ω1. We derive a necessary and sufficient condition in terms of trees for the existence of an U with $\omega_1 < \sigma(\mathfrak{U}) < 2^{\omega_1}$. We study the sufficiency of some conditions for σ(U) = 2ω1 . These conditions are analogous to conditions studied by D. Kueker in connection with countable models
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1027A taste of set theory for philosophersJournal of the Indian Council of Philosophical Research (2): 143-163. 2011.
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50Erratum to: On Definability in Dependence Logic (review)Journal of Logic, Language and Information 20 (1): 133-134. 2011.
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94On definability in dependence logicJournal of Logic, Language and Information 18 (3): 317-332. 2009.We study the expressive power of open formulas of dependence logic introduced in Väänänen [Dependence logic (Vol. 70 of London Mathematical Society Student Texts), 2007]. In particular, we answer a question raised by Wilfrid Hodges: how to characterize the sets of teams definable by means of identity only in dependence logic, or equivalently in independence friendly logic.
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43Aesthetics and the Dream of Objectivity: Notes from Set TheoryInquiry: An Interdisciplinary Journal of Philosophy 58 (1): 83-98. 2015.In this paper, we consider various ways in which aesthetic value bears on, if not serves as evidence for, the truth of independent statements in set theory.... the aesthetic issue, which in practice will also for me be the decisive factor—John von Neumann, letter to Carnap, 1931For me, it is the aesthetics which may very well be the final arbiter—P. J. Cohen, 2002
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12The hierarchy theorem for generalized quantifiersJournal of Symbolic Logic 61 (3): 802-817. 1996.The concept of a generalized quantifier of a given similarity type was defined in [12]. Our main result says that on finite structures different similarity types give rise to different classes of generalized quantifiers. More exactly, for every similarity typetthere is a generalized quantifier of typetwhich is not definable in the extension of first order logic by all generalized quantifiers of type smaller thant. This was proved for unary similarity types by Per Lindström [17] with a counting a…Read more
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9Stationary Sets and Infinitary LogicJournal of Symbolic Logic 65 (3): 1311-1320. 2000.Let K$^0_\lambda$ be the class of structures $\langle\lambda,
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61Second‐Order Logic and Set TheoryPhilosophy Compass 10 (7): 463-478. 2015.Both second-order logic and set theory can be used as a foundation for mathematics, that is, as a formal language in which propositions of mathematics can be expressed and proved. We take it upon ourselves in this paper to compare the two approaches, second-order logic on one hand and set theory on the other hand, evaluating their merits and weaknesses. We argue that we should think of first-order set theory as a very high-order logic
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49Partially ordered connectivesZeitschrift fur mathematische Logik und Grundlagen der Mathematik 38 (1): 361-372. 1992.We show that a coherent theory of partially ordered connectives can be developed along the same line as partially ordered quantification. We estimate the expressive power of various partially ordered connectives and use methods like Ehrenfeucht games and infinitary logic to get various undefinability results
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61Internal Categoricity in Arithmetic and Set TheoryNotre Dame Journal of Formal Logic 56 (1): 121-134. 2015.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…Read more
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