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Roman Kossak
City University of New York
  1.  8
    Automorphisms of Recursively Saturated Models of Arithmetic.Richard Kaye, Roman Kossak & Henryk Kotlarski - 1991 - Annals of Pure and Applied Logic 55 (1):67-99.
    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 classification (...)
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  2.  10
    Disjunctions with Stopping Conditions.Roman Kossak & Bartosz Wcisło - forthcoming - Bulletin of Symbolic Logic:1-28.
  3.  6
    On Maximal Subgroups of the Automorphism Group of a Countable Recursively Saturated Model of PA.Roman Kossak, Henryk Kotlarski & James H. Schmerl - 1993 - Annals of Pure and Applied Logic 65 (2):125-148.
    We show that the stabilizer of an element a of a countable recursively saturated model of arithmetic M is a maximal subgroup of Aut iff the type of a is selective. This is a point of departure for a more detailed study of the relationship between pointwise and setwise stabilizers of certain subsets of M and the types of elements in those subsets. We also show that a complete type of PA is 2-indiscernible iff it is minimal in the sense (...)
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  4.  12
    Arithmetically Saturated Models of Arithmetic.Roman Kossak & James H. Schmerl - 1995 - Notre Dame Journal of Formal Logic 36 (4):531-546.
    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 substructures.
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  5.  23
    On Two Questions Concerning the Automorphism Groups of Countable Recursively Saturated Models of PA.Roman Kossak & Nicholas Bamber - 1996 - Archive for Mathematical Logic 36 (1):73-79.
  6.  44
    A Certain Class of Models of Peano Arithmetic.Roman Kossak - 1983 - Journal of Symbolic Logic 48 (2):311-320.
  7.  12
    Minimal Satisfaction Classes with an Application to Rigid Models of Peano Arithmetic.Roman Kossak & James H. Schmerl - 1991 - Notre Dame Journal of Formal Logic 32 (3):392-398.
  8.  21
    A Note on Satisfaction Classes.Roman Kossak - 1985 - Notre Dame Journal of Formal Logic 26 (1):1-8.
  9.  15
    Contents.Andrés Villaveces, Roman Kossak, Juha Kontinen & Åsa Hirvonen - 2015 - 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.
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  10.  32
    Models with the Ω-Property.Roman Kossak - 1989 - Journal of Symbolic Logic 54 (1):177-189.
    A model M of PA has the omega-property if it has a subset of order type omega that is coded in an elementary end extension of M. All countable recursively saturated models have the omega-property, but there are also models with the omega-property that are not recursively saturated. The papers is devoted to the study of structural properties of such models.
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  11.  24
    Four Problems Concerning Recursively Saturated Models of Arithmetic.Roman Kossak - 1995 - Notre Dame Journal of Formal Logic 36 (4):519-530.
    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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  12.  13
    Recursively Saturated $\omega_1$-Like Models of Arithmetic.Roman Kossak - 1985 - Notre Dame Journal of Formal Logic 26 (4):413-422.
  13.  23
    Undefinability of Truth and Nonstandard Models.Roman Kossak - 2004 - Annals of Pure and Applied Logic 126 (1-3):115-123.
    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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  14.  56
    The Complexity of Classification Problems for Models of Arithmetic.Samuel Coskey & Roman Kossak - 2010 - Bulletin of Symbolic Logic 16 (3):345-358.
    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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  15.  11
    A Note on a Theorem of Kanovei.Roman Kossak - 2004 - Archive for Mathematical Logic 43 (4):565-569.
    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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  16.  16
    On Cofinal Submodels and Elementary Interstices.Roman Kossak & James H. Schmerl - 2012 - Notre Dame Journal of Formal Logic 53 (3):267-287.
    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 (...)
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  17.  22
    Subsets of Models of Arithmetic.Roman Kossak & Jeffrey B. Paris - 1992 - Archive for Mathematical Logic 32 (1):65-73.
    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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  18.  27
    Automorphism Group Actions on Trees.Alexandre Ivanov & Roman Kossak - 2004 - Mathematical Logic Quarterly 50 (1):71.
    We study the situation when the automorphism group of a recursively saturated structure acts on an ℝ-tree. The cases of and models of Peano Arithmetic are central in the paper.
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  19.  42
    A Note on BΣn and an Intermediate Induction Schema.Zofia Adamowicz & Roman Kossak - 1988 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 34 (3):261-264.
  20.  14
    Models with the $Omega$-Property.Roman Kossak - 1989 - Journal of Symbolic Logic 54 (1):177-189.
  21.  1
    Neutrally Expandable Models of Arithmetic.Athar Abdul‐Quader & Roman Kossak - 2019 - Mathematical Logic Quarterly 65 (2):212-217.
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  22.  8
    Logic Without Borders: Essays on Set Theory, Model Theory, Philosophical Logic and Philosophy of Mathematics.Andrés Villaveces, Roman Kossak, Juha Kontinen & Åsa Hirvonen (eds.) - 2015 - De Gruyter.
    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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  23.  14
    Logic Without Borders: Essays on Set Theory, Model Theory, Philosophical Logic and Philosophy of Mathematics.Åsa Hirvonen, Juha Kontinen, Roman Kossak & Andrés Villaveces (eds.) - 2015 - De Gruyter.
  24.  52
    Set Theory, Arithmetic, and Foundations of Mathematics: Theorems, Philosophies.Juliette Kennedy & Roman Kossak (eds.) - 2011 - Cambridge University Press.
    Machine generated contents note: 1. Introduction Juliette Kennedy and Roman Kossak; 2. Historical remarks on Suslin's problem Akihiro Kanamori; 3. The continuum hypothesis, the generic-multiverse of sets, and the [OMEGA] conjecture W. Hugh Woodin; 4. [omega]-Models of finite set theory Ali Enayat, James H. Schmerl and Albert Visser; 5. Tennenbaum's theorem for models of arithmetic Richard Kaye; 6. Hierarchies of subsystems of weak arithmetic Shahram Mohsenipour; 7. Diophantine correct open induction Sidney Raffer; 8. Tennenbaum's theorem and recursive reducts James H. (...)
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  25.  22
    A Note on the Multiplicative Semigroup of Models of Peano Arithmetic.Roman Kossak, Mark Nadel & James Schmerl - 1989 - Journal of Symbolic Logic 54 (3):936-940.
  26.  31
    Game Approximations of Satisfaction Classes Models.Roman Kossak & Henryk Kotlarski - 1992 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 38 (1):21-26.
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  27. James H. Schmerl. Peano Models with Many Generic Classes. Pacific Journal of Mathematics, Vol. 43 (1973), Pp. 523–536. - James H. Schmerl. Correction To: “Peano Models with Many Generic Classes”. Pacific Journal of Mathematics, Vol. 92 (1981), No. 1, Pp. 195–198. - James H. Schmerl. Recursively Saturated, Rather Classless Models of Peano Arithmetic. Logic Year 1979–80. Recursively Saturated, Rather Classless Models of Peano Arithmetic. Logic Year 1979–80 (Proceedings, Seminars, and Conferences in Mathematical Logic, University of Connecticut, Storrs, Connecticut, 1979/80). Edited by M. Lerman, J. H. Schmerl, and R. I. Soare, Lecture Notes in Mathematics, Vol. 859. Springer, Berlin, Pp. 268–282. - James H. Schmerl. Recursively Saturatedmodels Generated by Indiscernibles. Notre Dane Journal of Formal Logic, Vol. 26 (1985), No. 1, Pp. 99–105. - James H. Schmerl. Large Resplendent Models Generated by Indiscernibles. The Journal of Symbolic Logic, Vol. 54 (1989), No. 4, Pp. 1382–1388. - Jam. [REVIEW]Roman Kossak - 2009 - Bulletin of Symbolic Logic 15 (2):222-227.
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  28.  9
    Logic & Structure: An Art Project.Roman Kossak & Wanda Siedlecka - 2021 - Theoria 87 (4):959-970.
    Theoria, Volume 87, Issue 4, Page 959-970, August 2021.
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  29.  10
    Open Days in Set Theory and Arithmetic, Jachranka, Poland, 1986.Roman Kossak & Marian Srebrny - 1987 - Journal of Symbolic Logic 52 (3):888-894.
  30.  17
    The Ω-Like Recursively Saturated Models of Arithmetic.Roman Kossak - 1991 - Bulletin of the Section of Logic 20 (3/4):109-109.
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  31.  17
    The Notre Dame Lectures, Edited by Peter Cholak, Lecture Notes in Logic, Vol. 18. Association for Symbolic Logic, A K Peters, Ltd., Wellesley, Massachusetts, 2005, Vii + 185 Pp. [REVIEW]Roman Kossak - 2006 - Bulletin of Symbolic Logic 12 (4):605-607.
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  32. The Structure of Models of Peano Arithmetic.Roman Kossak & James Schmerl - 2006 - Oxford, England: Clarendon Press.
    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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  33.  6
    A Radio Interview with Jouko Väänänen.Andrés Villaveces, Roman Kossak, Juha Kontinen & Åsa Hirvonen - 2015 - 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. pp. 417-422.
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  34.  7
    Preface – Unity and Diversity of Logic.Andrés Villaveces, Roman Kossak, Juha Kontinen & Åsa Hirvonen - 2015 - 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.
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