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1022A 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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93On 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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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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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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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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22Trees and $Pi^11$-Subsets of $^{omega_1}omega1$Journal of Symbolic Logic 58 (3): 1052-1070. 1993.We study descriptive set theory in the space $^{\omega_1}\omega_1$ 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 $\Pi^1_1$-sets of $^{\omega_1}\omega_1$. We call a family $\mathscr{U}$ of trees universal for a class $\mathscr{V}$ of trees if $\mathscr{U} \subseteq \mathscr{V}$ and every tree in $\mathscr{V}$ can be order-preservingly mapp…Read more
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14Boolean valued models and generalized quantifiersAnnals of Mathematical Logic 18 (3): 193-225. 1980.
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59Axiomatizing first-order consequences in dependence logicAnnals of Pure and Applied Logic 164 (11): 1101-1117. 2013.Dependence logic, introduced in Väänänen [11], cannot be axiomatized. However, first-order consequences of dependence logic sentences can be axiomatized, and this is what we shall do in this paper. We give an explicit axiomatization and prove the respective Completeness Theorem
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