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91Arithmetically Saturated Models of ArithmeticNotre Dame Journal of Formal Logic 36 (4): 531-546. 1995.The paper presents an outline of the general theory of countable arithmetically saturated models of PA and some of its applications. We consider questions concerning the automorphism group of a countable recursively saturated model of PA. We prove new results concerning fixed point sets, open subgroups, and the cofinality of the automorphism group. We also prove that the standard system of a countable arithmetically saturated model of PA is determined by the lattice of its elementary substructur…Read more
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76On two questions concerning the automorphism groups of countable recursively saturated models of PAArchive for Mathematical Logic 36 (1): 73-79. 1996.
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56A note on a theorem of KanoveiArchive for Mathematical Logic 43 (4): 565-569. 2004.We give a short proof of a theorem of Kanovei on separating induction and collection schemes for Σ n formulas using families of subsets of countable models of arithmetic coded in elementary end extensions
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101A Note on BΣn and an Intermediate Induction SchemaZeitschrift fur mathematische Logik und Grundlagen der Mathematik 34 (3): 261-264. 1988.
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48A Radio Interview with Jouko VäänänenIn 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. 417-422. 2015.
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36The Structure of Models of Peano ArithmeticClarendon Press. 2006.Aimed at graduate students, research logicians and mathematicians, this much-awaited text covers over 40 years of work on relative classification theory for nonstandard models of arithmetic. The book covers basic isomorphism invariants: families of type realized in a model, lattices of elementary substructures and automorphism groups.
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87On Cofinal Submodels and Elementary IntersticesNotre Dame Journal of Formal Logic 53 (3): 267-287. 2012.We prove a number of results concerning the variety of first-order theories and isomorphism types of pairs of the form $(N,M)$ , where $N$ is a countable recursively saturated model of Peano Arithmetic and $M$ is its cofinal submodel. We identify two new isomorphism invariants for such pairs. In the strongest result we obtain continuum many theories of such pairs with the fixed greatest common initial segment of $N$ and $M$ and fixed lattice of interstructures $K$ , such that $M\prec K\prec N$
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87Automorphisms of recursively saturated models of arithmeticAnnals of Pure and Applied Logic 55 (1): 67-99. 1991.We give an examination of the automorphism group Aut of a countable recursively saturated model M of PA. The main result is a characterisation of strong elementary initial segments of M as the initial segments consisting of fixed points of automorphisms of M. As a corollary we prove that, for any consistent completion T of PA, there are recursively saturated countable models M1, M2 of T, such that Aut[ncong]Aut, as topological groups with a natural topology. Other results include a classificatio…Read more
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83Four Problems Concerning Recursively Saturated Models of ArithmeticNotre Dame Journal of Formal Logic 36 (4): 519-530. 1995.The paper presents four open problems concerning recursively saturated models of Peano Arithmetic. One problems concerns a possible converse to Tarski's undefinability of truth theorem. The other concern elementary cuts in countable recursively saturated models, extending automorphisms of countable recursively saturated models, and Jonsson models of PA. Some partial answers are given.
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82Subsets of models of arithmeticArchive for Mathematical Logic 32 (1): 65-73. 1992.We define certain properties of subsets of models of arithmetic related to their codability in end extensions and elementary end extensions. We characterize these properties using some more familiar notions concerning cuts in models of arithmetic
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30Automorphisms of Recursively Saturated Models of Peano Arithmetic: Fixed Point SetsLogic Journal of the IGPL 5 (6): 787-794. 1997.We consider the question: If M is a countable recursively saturated model of PA and K is an elementary submodel of M, is there an automorphism α of M such that K is the fixed point set of α? We give a survey of the known results and we prove that, if M is arithmetically saturated, then M has continuum many pairwise nonisomorphic elementary submodels which are fixed point sets
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230The complexity of classification problems for models of arithmeticBulletin of Symbolic Logic 16 (3): 345-358. 2010.We observe that the classification problem for countable models of arithmetic is Borel complete. On the other hand, the classification problems for finitely generated models of arithmetic and for recursively saturated models of arithmetic are Borel; we investigate the precise complexity of each of these. Finally, we show that the classification problem for pairs of recursively saturated models and for automorphisms of a fixed recursively saturated model are Borel complete.
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66ContentsIn 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. 2015.
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91Undefinability of truth and nonstandard modelsAnnals of Pure and Applied Logic 126 (1-3): 115-123. 2004.We discuss Robinson's model theoretic proof of Tarski's theorem on undefinability of truth. We present two other “diagonal-free” proofs of Tarski's theorem, and we compare undefinability of truth to other forms of undefinability in nonstandard models of arithmetic
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78Open days in set theory and arithmetic, Jachranka, Poland, 1986Journal of Symbolic Logic 52 (3): 888-894. 1987.
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82Minimal satisfaction classes with an application to rigid models of Peano arithmeticNotre Dame Journal of Formal Logic 32 (3): 392-398. 1991.
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59The ω-like recursively saturated models of arithmeticBulletin of the Section of Logic 20 (3/4): 109-109. 1991.
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83Game approximations of satisfaction classes modelsZeitschrift fur mathematische Logik und Grundlagen der Mathematik 38 (1): 21-26. 1992.
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150Logic Without Borders: Essays on Set Theory, Model Theory, Philosophical Logic and Philosophy of Mathematics (edited book)De Gruyter. 2015.In recent years, mathematical logic has developed in many directions, the initial unity of its subject matter giving way to a myriad of seemingly unrelated areas. The articles collected here, which range from historical scholarship to recent research in geometric model theory, squarely address this development. These articles also connect to the diverse work of Väänänen, whose ecumenical approach to logic reflects the unity of the discipline.
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56Preface – Unity and Diversity of LogicIn 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. 2015.
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CUNY Graduate CenterRegular Faculty
New York City, New York, United States of America
Areas of Interest
| Logic and Philosophy of Logic |
| European Philosophy |