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Profile: Jouko Väänänen (University of Helsinki, University of Amsterdam)
  1.  7
    Erich Grädel & Jouko Väänänen (2013). Dependence and Independence. Studia Logica 101 (2):399-410.
    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 ${\vec{y} \bot_{\vec{x}}\vec{z}}$ gives rise to a natural logic capable of formalizing basic intuitions about independence and dependence. (...)
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  2.  53
    Jouko Väänänen (2012). Second Order Logic or Set Theory? Bulletin of Symbolic Logic 18 (1):91-121.
    We try to answer the question which is the “right” foundation of mathematics, second order logic or set theory. Since the former is usually thought of as a formal language and the latter as a first order theory, we have to rephrase the question. We formulate what we call the second order view and a competing set theory view, and then discuss the merits of both views. On the surface these two views seem to be in manifest conflict with each (...)
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  3.  31
    Juha Kontinen & Jouko Väänänen (2009). On Definability in Dependence Logic. Journal of Logic, Language and Information 18 (3):317-332.
    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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  4.  39
    Samson Abramsky & Jouko Väänänen (2009). From If to Bi. Synthese 167 (2):207 - 230.
    We take a fresh look at the logics of informational dependence and independence of Hintikka and Sandu and Väänänen, and their compositional semantics due to Hodges. We show how Hodges’ semantics can be seen as a special case of a general construction, which provides a context for a useful completeness theorem with respect to a wider class of models. We shed some new light on each aspect of the logic. We show that the natural propositional logic carried by the semantics (...)
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  5.  59
    Jouko Vaananen (2001). Second-Order Logic and Foundations of Mathematics. Bulletin of Symbolic Logic 7 (4):504-520.
    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 set theory and second-order logic are not radically (...)
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  6.  8
    Jouko Väänänen & Tong Wang (2015). Internal Categoricity in Arithmetic and Set Theory. Notre Dame Journal of Formal Logic 56 (1):121-134.
    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 is, the question of existence of (...)
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  7.  7
    Daisuke Ikegami & Jouko Väänänen (2015). Boolean-Valued Second-Order Logic. Notre Dame Journal of Formal Logic 56 (1):167-190.
    In so-called full second-order logic, the second-order variables range over all subsets and relations of the domain in question. In so-called Henkin second-order logic, every model is endowed with a set of subsets and relations which will serve as the range of the second-order variables. In our Boolean-valued second-order logic, the second-order variables range over all Boolean-valued subsets and relations on the domain. We show that under large cardinal assumptions Boolean-valued second-order logic is more robust than full second-order logic. Its (...)
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  8.  8
    Juha Kontinen & Jouko Väänänen (2013). Axiomatizing First-Order Consequences in Dependence Logic. Annals of Pure and Applied Logic 164 (11):1101-1117.
    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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  9.  16
    Juha Kontinen & Jouko Väänänen (2010). A Remark on Negation in Dependence Logic. Notre Dame Journal of Formal Logic 52 (1):55-65.
    We show that for any pair $\phi$ and $\psi$ of contradictory formulas of dependence logic there is a formula $\theta$ of the same logic such that $\phi\equiv\theta$ and $\psi\equiv\neg\theta$. This generalizes a result of Burgess.
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  10. Lauri Hella, Kerkko Luosto & Jouko Väänänen (1996). The Hierarchy Theorem for Generalized Quantifiers. Journal of Symbolic Logic 61 (3):802-817.
    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 (...)
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  11.  83
    Jouko Väänänen (2011). A Taste of Set Theory for Philosophers. Journal of the Indian Council of Philosophical Research (2):143-163.
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  12. Jouko Väänänen (2002). On the Semantics of Informational Independence. Logic Journal of the Igpl 10 (3):339-352.
    The semantics of the independence friendly logic of Hintikka and Sandu is usually defined via a game of imperfect information. We give a definition in terms of a game of perfect information. We also give an Ehrenfeucht-Fraïssé game adequate for this logic and use it to define a Distributive Normal Form for independence friendly logic.
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  13.  8
    Jouko Väänänen & Wilfrid Hodges (2010). Dependence of Variables Construed as an Atomic Formula. Annals of Pure and Applied Logic 161 (6):817-828.
    We define a logic capable of expressing dependence of a variable on designated variables only. Thus has similar goals to the Henkin quantifiers of [4] and the independence friendly logic of [6] that it much resembles. The logic achieves these goals by realizing the desired dependence declarations of variables on the level of atomic formulas. By [3] and [17], ability to limit dependence relations between variables leads to existential second order expressive power. Our avoids some difficulties arising in the original (...)
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  14. Saharon Shelah, Heikki Tuuri & Jouko Väänänen (1993). On the Number of Automorphisms of Uncountable Models. Journal of Symbolic Logic 58 (4):1402-1418.
    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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  15.  22
    Lauri Hella, Jouko Väänänen & Dag Westerståhl (1997). Definability of Polyadic Lifts of Generalized Quantifiers. Journal of Logic, Language and Information 6 (3):305-335.
    We study generalized quantifiers on finite structures.With every function : we associate a quantifier Q by letting Q x say there are at least (n) elementsx satisfying , where n is the sizeof the universe. This is the general form ofwhat is known as a monotone quantifier of type .We study so called polyadic liftsof such quantifiers. The particular lifts we considerare Ramseyfication, branching and resumption.In each case we get exact criteria fordefinability of the lift in terms (...)
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  16.  4
    Gabriel Sandu & Jouko Väänänen (1992). Partially Ordered Connectives. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 38 (1):361-372.
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  17.  6
    Saharon Shelah & Jouko Väänänen (2000). Stationary Sets and Infinitary Logic. Journal of Symbolic Logic 65 (3):1311-1320.
    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 (...)
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  18.  8
    Jouko Väänänen (2015). Second‐Order Logic and Set Theory. Philosophy Compass 10 (7):463-478.
    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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  19.  5
    Saharon Shelah & Jouko Väänänen (2005). A Note on Extensions of Infinitary Logic. Archive for Mathematical Logic 44 (1):63-69.
    We show that a strong form of the so called Lindström’s Theorem [4] fails to generalize to extensions of L κ ω and L κ κ : For weakly compact κ there is no strongest extension of L κ ω with the (κ,κ)-compactness property and the Löwenheim-Skolem theorem down to κ. With an additional set-theoretic assumption, there is no strongest extension of L κ κ with the (κ,κ)-compactness property and the Löwenheim-Skolem theorem down to <κ.
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  20.  14
    Jouko Väänänen (2015). Categoricity and Consistency in Second-Order Logic. Inquiry 58 (1):20-27.
    We analyse the concept of a second-order characterisable structure and divide this concept into two parts—consistency and categoricity—with different strength and nature. We argue that categorical characterisation of mathematical structures in second-order logic is meaningful and possible without assuming that the semantics of second-order logic is defined in set theory. This extends also to the so-called Henkin structures.
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  21.  27
    Jouko Väänänen & Dag Westerståhl (2002). On the Expressive Power of Monotone Natural Language Quantifiers Over Finite Models. Journal of Philosophical Logic 31 (4):327-358.
    We study definability in terms of monotone generalized quantifiers satisfying Isomorphism Closure, Conservativity and Extension. Among the quantifiers with the latter three properties - here called CE quantifiers - one finds the interpretations of determiner phrases in natural languages. The property of monotonicity is also linguistically ubiquitous, though some determiners like an even number of are highly non-monotone. They are nevertheless definable in terms of monotone CE quantifiers: we give a necessary and sufficient condition for such definability. We further identify (...)
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  22.  15
    Jouko Väänänen (1997). Unary Quantifiers on Finite Models. Journal of Logic, Language and Information 6 (3):275-304.
    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 of agiven monotone quantifier.Our analysis shows that (...)
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  23.  10
    Jouko Väänänen (1995). Games and Trees in Infinitary Logic: A Survey. In M. Krynicki, M. Mostowski & L. Szczerba (eds.), Quantifiers: Logics, Models and Computation. Kluwer Academic Publishers 105--138.
  24.  9
    Michael Krynicki & Jouko Väänänen (1989). Henkin and Function Quantifiers. Annals of Pure and Applied Logic 43 (3):273-292.
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  25.  1
    Peter Aczel, Seppo Miettinen & Jouko Vaananen (1984). The Strength of Martin-Löf's Intuitionistic Type Theory with One Universe. Journal of Symbolic Logic 49 (1):313-313.
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  26.  6
    Heinrich Herre, Michał Krynicki, Alexandr Pinus & Jouko Väänänen (1991). The Härtig Quantifier: A Survey. Journal of Symbolic Logic 56 (4):1153-1183.
    A fundamental notion in a large part of mathematics is the notion of equicardinality. The language with Hartig quantifier is, roughly speaking, a first-order language in which the notion of equicardinality is expressible. Thus this language, denoted by LI, is in some sense very natural and has in consequence special interest. Properties of LI are studied in many papers. In [BF, Chapter VI] there is a short survey of some known results about LI. We feel that a more extensive exposition (...)
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  27.  20
    Jouko Väänänen (2008). The Craig Interpolation Theorem in Abstract Model Theory. Synthese 164 (3):401 - 420.
    The Craig Interpolation Theorem is intimately connected with the emergence of abstract logic and continues to be the driving force of the field. I will argue in this paper that the interpolation property is an important litmus test in abstract model theory for identifying “natural,” robust extensions of first order logic. My argument is supported by the observation that logics which satisfy the interpolation property usually also satisfy a Lindström type maximality theorem. Admittedly, the range of such logics is small.
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  28.  4
    Gabriel Sandu & Jouko Väänänen (1992). Partially Ordered Connectives. Mathematical Logic Quarterly 38 (1):361-372.
    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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  29.  7
    Juliette Kennedy, Saharon Shelah & Jouko Väänänen (2008). Regular Ultrafilters and Finite Square Principles. Journal of Symbolic Logic 73 (3):817-823.
    We show that many singular cardinals λ above a strongly compact cardinal have regular ultrafilters D that violate the finite square principle $\square _{\lambda ,D}^{\mathit{fin}}$ introduced in [3]. For such ultrafilters D and cardinals λ there are models of size λ for which Mλ / D is not λ⁺⁺-universal and elementarily equivalent models M and N of size λ for which Mλ / D and Nλ / D are non-isomorphic. The question of the existence of such ultrafilters and models was (...)
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  30.  11
    Tapani Hyttinen & Jouko Väänänen (1990). On Scott and Karp Trees of Uncountable Models. 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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  31.  4
    Juliette Kennedy, Saharon Shelah & Jouko Väänänen (2015). Regular Ultrapowers at Regular Cardinals. 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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  32.  20
    Jouko Väänänen (2004). Barwise: Abstract Model Theory and Generalized Quantifiers. Bulletin of Symbolic Logic 10 (1):37-53.
  33.  11
    Jouko Vaananen (2004). Barwise: Abstract Model Theory and Generalized Quantifiers. Bulletin of Symbolic Logic 10 (1):37-53.
  34.  8
    Juliette Kennedy & Jouko Väänänen (2015). Aesthetics and the Dream of Objectivity: Notes From Set Theory. Inquiry 58 (1):83-98.
    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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  35.  11
    Alan Mekler & Jouko Vaananen (1993). Trees and Π11-Subsets of Ω1ω. Journal of Symbolic Logic 58 (3).
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  36.  1
    Michał Krynicki, Alistair Lachlan & Jouko Väänänen (1984). Vector Spaces and Binary Quantifiers. Notre Dame Journal of Formal Logic 25 (1):72-78.
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  37.  38
    Jouko Väänänen & Dag Westerståhl (2010). In memoriam: Per Lindström. Theoria 76 (2):100-107.
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  38.  29
    Menachem Magidor & Jouko Väänänen (2011). On Löwenheim–Skolem–Tarski Numbers for Extensions of First Order Logic. Journal of Mathematical Logic 11 (01):87-113.
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  39. France Xii, Marcelo Coniglio, Gilles Dowek, Jouko Väänanen, Renata Wassermann, Eric Allender, Jean-Baptiste Joinet & Dale Miller (2004). Ouro Preto (Minas Gerais), Brazil July 29–August 1, 2003. Bulletin of Symbolic Logic 10 (2).
     
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  40.  4
    Stevo Todorčević & Jouko Väänänen (1999). Trees and Ehrenfeucht–Fraı̈ssé Games. Annals of Pure and Applied Logic 100 (1-3):69-97.
    Trees are natural generalizations of ordinals and this is especially apparent when one tries to find an uncountable analogue of the concept of the Scott-rank of a countable structure. The purpose of this paper is to introduce new methods in the study of an ordering between trees whose analogue is the usual ordering between ordinals. For example, one of the methods is the tree-analogue of the successor operation on the ordinals.
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  41. Jouko Väänänen (2006). A Remark on Nondeterminacy in IF Logic. Acta Philosophica Fennica 78 (2006):71-77.
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  42.  2
    Tapani Hyttinen, Gianluca Paolini & Jouko Väänänen (2015). Quantum Team Logic and Bell’s Inequalities. 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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  43.  32
    Jouko Väänänen (1982). Abstract Logic and Set Theory. II. Large Cardinals. Journal of Symbolic Logic 47 (2):335-346.
    The following problem is studied: How large and how small can the Löwenheim and Hanf numbers of unbounded logics be in relation to the most common large cardinals? The main result is that the Löwenheim number of the logic with the Härtig-quantifier can be consistently put in between any two of the first weakly inaccessible, the first weakly Mahlo, the first weakly compact, the first Ramsey, the first measurable and the first supercompact cardinals.
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  44.  26
    Jouko Väänänen (2011). Erratum To: On Definability in Dependence Logic. [REVIEW] Journal of Logic, Language and Information 20 (1):133-134.
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  45.  14
    Juha Kontinen, Jouko Väänänen & Dag Westerståhl (2013). Editorial Introduction. Studia Logica 101 (2):233-236.
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  46.  1
    Phokion G. Kolaitis & Jouko A. Väänänen (1995). Generalized Quantifiers and Pebble Games on Finite Structures. Annals of Pure and Applied Logic 74 (1):23-75.
    First-order logic is known to have a severely limited expressive power on finite structures. As a result, several different extensions have been investigated, including fragments of second-order logic, fixpoint logic, and the infinitary logic L∞ωω in which every formula has only a finite number of variables. In this paper, we study generalized quantifiers in the realm of finite structures and combine them with the infinitary logic L∞ωω to obtain the logics L∞ωω, where Q = {Qi: iε I} is (...)
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  47.  6
    Jouko Väänänen & Boban Veličković (2004). Games Played on Partial Isomorphisms. Archive for Mathematical Logic 43 (1):19-30.
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  48.  6
    Alan Mekler & Jouko Vaananen (1993). Trees and $Pi^11$-Subsets of $^{Omega_1}Omega1$. Journal of Symbolic Logic 58 (3):1052-1070.
    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 mapped into a tree in $\mathscr{U}$. It is well (...)
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  49.  1
    Jouko Väänänen (2015). On Second Order Logic. Philosophical Inquiry 39 (1):59-62.
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  50.  1
    Jouko Väänänen & Lauri Hella (2015). The Size of a Formula as a Measure of Complexity. In Andrés Villaveces, Roman Kossak, Juha Kontinen & Åsa Hirvonen (eds.), Logic Without Borders: Essays on Set Theory, Model Theory, Philosophical Logic and Philosophy of Mathematics. De Gruyter 193-214.
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