30 found
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  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.  55
    On Interstices of Countable Arithmetically Saturated Models of Peano Arithmetic.Nicholas Bamber & Henryk Kotlarski - 1997 - Mathematical Logic Quarterly 43 (4):525-540.
    We give some information about the action of Aut on M, where M is a countable arithmetically saturated model of Peano Arithmetic. We concentrate on analogues of moving gaps and covering gaps inside M.
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  3.  23
    The Incompleteness Theorems After 70 Years.Henryk Kotlarski - 2004 - Annals of Pure and Applied Logic 126 (1-3):125-138.
    We give some information about new proofs of the incompleteness theorems, found in 1990s. Some of them do not require the diagonal lemma as a method of construction of an independent statement.
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  4.  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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  5.  13
    Full Satisfaction Classes: A Survey.Henryk Kotlarski - 1991 - Notre Dame Journal of Formal Logic 32 (4):573-579.
  6.  38
    On the Incompleteness Theorems.Henryk Kotlarski - 1994 - Journal of Symbolic Logic 59 (4):1414-1419.
    We give new proofs of both incompleteness theorems. We do not use the diagonalization lemma, but work with some quickly growing functions instead.
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  7.  17
    Other Proofs of Old Results.Henryk Kotlarski - 1998 - Mathematical Logic Quarterly 44 (4):474-480.
    We transform the proof of the second incompleteness theorem given in [3] to a proof-theoretic version, avoiding the use of the arithmetized completeness theorem. We give also new proofs of old results: The Arithmetical Hierarchy Theorem and Tarski's Theorem on undefinability of truth; the proofs in which the construction of a sentence by means of diagonalization lemma is not needed.
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  8. Some Remarks on Initial Segments in Models of Peano Arithmetic.Henryk Kotlarski - 1984 - Journal of Symbolic Logic 49 (3):955-960.
    If $M \models PA (= Peano Arithmetic)$ , we set $A^M = \{N \subset_e M: N \models PA\}$ and study this family.
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  9.  15
    On Cofinal Extensions of Models of Arithmetic.Henryk Kotlarski - 1983 - Journal of Symbolic Logic 48 (2):253-262.
    We study cofinal extensions of models of arithmetic, in particular we show that some properties near to expandability are preserved under cofinal extensions.
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  10.  14
    Automorphisms of Models of True Arithmetic: Recognizing Some Basic Open Subgroups.Henryk Kotlarski & Richard Kaye - 1994 - Notre Dame Journal of Formal Logic 35 (1):1-14.
    Let M be a countable recursively saturated model of Th(), and let GAut(M), considered as a topological group. We examine connections between initial segments of M and subgroups of G. In particular, for each of the following classes of subgroups HG, we give characterizations of the class of terms of the topological group structure of H as a subgroup of G. (a) for some (b) for some (c) for some (d) for some (Here, M(a) denotes the smallest M containing a, (...)
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  11.  11
    More on Lower Bounds for Partitioning Α-Large Sets.Henryk Kotlarski, Bożena Piekart & Andreas Weiermann - 2007 - Annals of Pure and Applied Logic 147 (3):113-126.
    Continuing the earlier research from [T. Bigorajska, H. Kotlarski, Partitioning α-large sets: some lower bounds, Trans. Amer. Math. Soc. 358 4981–5001] we show that for the price of multiplying the number of parts by 3 we may construct partitions all of whose homogeneous sets are much smaller than in [T. Bigorajska, H. Kotlarski, Partitioning α-large sets: some lower bounds, Trans. Amer. Math. Soc. 358 4981–5001]. We also show that the Paris–Harrington independent statement remains unprovable if the number of colors is (...)
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  12.  29
    On Models Constructed by Means of the Arithmetized Completeness Theorem.Richard Kaye & Henryk Kotlarski - 2000 - Mathematical Logic Quarterly 46 (4):505-516.
    In this paper we study the model theory of extensions of models of first-order Peano Arithmetic by means of the arithmetized completeness theorem applied to a definable complete extension of PA in the original model. This leads us to many interesting model theoretic properties equivalent to reflection principles and ω-consistency, and these properties together with the associated first-order schemes extending PA are studied.
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  13.  32
    An Addition to Rosser's Theorem.Henryk Kotlarski - 1996 - Journal of Symbolic Logic 61 (1):285-292.
    For a primitive recursive consistent and strong enough theory T we construct an independent statement which has some clear metamathematical meaning.
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  14.  39
    Automorphisms of Countable Recursively Saturated Models of PA: A Survey.Henryk Kotlarski - 1995 - Notre Dame Journal of Formal Logic 36 (4):505-518.
    We give a survey of automorphisms of countable recursively saturated models of Peano Arithmetic.
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  15.  16
    Automorphisms of Countable Recursively Saturated Models of PA: Open Subgroups and Invariant Cuts.Henryk Kotlarski & Bozena Piekart - 1995 - Mathematical Logic Quarterly 41 (1):138-142.
    Let M be a countable recursively saturated model of PA and H an open subgroup of G = Aut. We prove that I = sup {b ∈ M : ∀u < bfu = u and J = inf{b ∈ MH} may be invariant, i. e. fixed by all automorphisms of M.
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  16.  32
    Bounded Induction and Satisfaction Classes.Henryk Kotlarski - 1986 - Mathematical Logic Quarterly 32 (31-34):531-544.
  17.  5
    Automorphisms of Models of True Arithmetic: Subgroups Which Extend to a Maximal Subgroup Uniquely.Henryk Kotlarski & Bożena Piekart - 1994 - Mathematical Logic Quarterly 40 (1):95-102.
    We show that if M is a countable recursively saturated model of True Arithmetic, then G = Aut has nonmaximal open subgroups with unique extension to a maximal subgroup of Aut.
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  18.  34
    More on Induction in the Language with a Satisfaction Class.Henryk Kotlarski & Zygmunt Ratajczyk - 1990 - Mathematical Logic Quarterly 36 (5):441-454.
  19.  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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  20.  7
    A. D. Tajmanov. Haraktéristiki Aksiomatiziruémyh Klassov Modélej . Algébra I Logika, Séminar, Vol. 1 No. 4 , Pp. 5–31.Henryk Kotlarski - 1973 - Journal of Symbolic Logic 38 (1):164.
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  21. A Model–Theoretic Approach to Proof Theory.Henryk Kotlarski - 2019 - Springer Verlag.
    This book presents a detailed treatment of ordinal combinatorics of large sets tailored for independence results. It uses model theoretic and combinatorial methods to obtain results in proof theory, such as incompleteness theorems or a description of the provably total functions of a theory. In the first chapter, the authors first discusses ordinal combinatorics of finite sets in the style of Ketonen and Solovay. This provides a background for an analysis of subsystems of Peano Arithmetic as well as for combinatorial (...)
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  22.  5
    Automorphisms of Models of True Arithmetic: More on Subgroups Which Extend to a Maximal One Uniquely.Henryk Kotlarski & Bożena Piekart - 2000 - Mathematical Logic Quarterly 46 (1):111-120.
    Continuing the earlier research in [14] we give some more information about nonmaximal open subgroups of G = Aut with unique maximal extension, where ℳ is a countable recursively saturated model of True Arithmetic.
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  23.  23
    On a Question of Andreas Weiermann.Henryk Kotlarski & Konrad Zdanowski - 2009 - Mathematical Logic Quarterly 55 (2):201-211.
    We prove that for each β, γ < ε0 there existsα < ε0 such that whenever A ⊆ ω is α -large and G: A → β is such that ) ≤ a), then there exists a γ -large C ⊆ A on which G is nondecreasing. Moreover, we give upper bounds for α for small ordinals β ≤ ωmath image.
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  24.  7
    On the End Extension Problem For Δ0‐PA(S).Henryk Kotlarski - 1989 - Mathematical Logic Quarterly 35 (5):391-397.
  25.  21
    On the End Extension Problem For Δ0-PA.Henryk Kotlarski - 1989 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 35 (5):391-397.
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  26.  2
    Review: A. D. Tajmanov, Characteristics of Axiomatizable Classes of Models. [REVIEW]Henryk Kotlarski - 1973 - Journal of Symbolic Logic 38 (1):164-164.
  27.  18
    Some Variations of the Hardy Hierarchy.Henryk Kotlarski - 2005 - Mathematical Logic Quarterly 51 (4):417.
    We study some variations of the so-called Hardy hierarchy of quickly growing functions, known from the literature, and obtain analogues of Ratajczyk's approximation lemma for them.
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  28.  6
    The Recursively Saturated Part of Models of Peano Arithmetic.Henryk Kotlarski - 1986 - Mathematical Logic Quarterly 32 (19‐24):365-370.
  29.  17
    The Recursively Saturated Part of Models of Peano Arithmetic.Henryk Kotlarski - 1986 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 32 (19-24):365-370.
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  30.  22
    On Skolem Ultrapowers and Their Non-Standard Variant.Henryk Kotlarski - 1980 - Mathematical Logic Quarterly 26 (14-18):227-236.