63 found
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  1.  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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  2.  45
    Saturation and Simple Extensions of Models of Peano Arithmetic.Matt Kaufmann & James H. Schmerl - 1984 - Annals of Pure and Applied Logic 27 (2):109-136.
  3.  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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  4.  25
    On the Role of Ramsey Quantifiers in First Order Arithmetic.James H. Schmerl & Stephen G. Simpson - 1982 - Journal of Symbolic Logic 47 (2):423-435.
  5.  13
    Subsets Coded in Elementary End Extensions.James H. Schmerl - 2014 - Archive for Mathematical Logic 53 (5-6):571-581.
  6.  5
    Recursively Saturated Models Generated by Indiscernibles.James H. Schmerl - 1985 - Notre Dame Journal of Formal Logic 26 (2):99-105.
  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.  32
    Making the Hyperreal Line Both Saturated and Complete.H. Jerome Keisler & James H. Schmerl - 1991 - Journal of Symbolic Logic 56 (3):1016-1025.
    In a nonstandard universe, the κ-saturation property states that any family of fewer than κ internal sets with the finite intersection property has a nonempty intersection. An ordered field F is said to have the λ-Bolzano-Weierstrass property iff F has cofinality λ and every bounded λ-sequence in F has a convergent λ-subsequence. We show that if $\kappa < \lambda$ are uncountable regular cardinals and $\beta^\alpha < \lambda$ whenever $\alpha < \kappa$ and $\beta < \lambda$, then there is a κ-saturated nonstandard (...)
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  9.  5
    On Κ-Like Structures Which Embed Stationary and Closed Unbounded Subsets.James H. Schmerl - 1976 - Annals of Mathematical Logic 10 (3-4):289-314.
  10.  21
    Remarks on Weak Notions of Saturation in Models of Peano Arithmetic.Matt Kaufmann & James H. Schmerl - 1987 - Journal of Symbolic Logic 52 (1):129-148.
  11.  74
    The Isomorphism Property for Nonstandard Universes.James H. Schmerl - 1995 - Journal of Symbolic Logic 60 (2):512-516.
  12.  12
    Making the Hyperreal Line Both Saturated and Complete.H. Jerome Keisler & James H. Schmerl - 1991 - Journal of Symbolic Logic 56 (3):1016-1025.
    In a nonstandard universe, the $\kappa$-saturation property states that any family of fewer than $\kappa$ internal sets with the finite intersection property has a nonempty intersection. An ordered field $F$ is said to have the $\lambda$-Bolzano-Weierstrass property iff $F$ has cofinality $\lambda$ and every bounded $\lambda$-sequence in $F$ has a convergent $\lambda$-subsequence. We show that if $\kappa < \lambda$ are uncountable regular cardinals and $\beta^\alpha < \lambda$ whenever $\alpha < \kappa$ and $\beta < \lambda$, then there is a $\kappa$-saturated nonstandard (...)
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  13.  10
    Moving Intersticial Gaps.James H. Schmerl - 2002 - Mathematical Logic Quarterly 48 (2):283-296.
    In a countable, recursively saturated model of Peano Arithmetic, an interstice is a maximal convex set which does not contain any definable elements. The interstices are partitioned into intersticial gaps in a way that generalizes the partition of the unbounded interstice into gaps. Continuing work of Bamber and Kotlarski [1], we investigate extensions of Kotlarski's Moving Gaps Lemma to the moving of intersticial gaps.
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  14.  9
    On Power-Like Models for Hyperinaccessible Cardinals.James H. Schmerl & Saharon Shelah - 1972 - Journal of Symbolic Logic 37 (3):531-537.
  15.  18
    Elementary Cuts in Saturated Models of Peano Arithmetic.James H. Schmerl - 2012 - Notre Dame Journal of Formal Logic 53 (1):1-13.
    A model $\mathscr{M} = (M,+,\times, 0,1,<)$ of Peano Arithmetic $({\sf PA})$ is boundedly saturated if and only if it has a saturated elementary end extension $\mathscr{N}$. The ordertypes of boundedly saturated models of $({\sf PA})$ are characterized and the number of models having these ordertypes is determined. Pairs $(\mathscr{N},M)$, where $\mathscr{M} \prec_{\sf end} \mathscr{N} \models({\sf PA})$ for saturated $\mathscr{N}$, and their theories are investigated. One result is: If $\mathscr{N}$ is a $\kappa$-saturated model of $({\sf PA})$ and $\mathscr{M}_0, \mathscr{M}_1 \prec_{\sf end} (...)
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  16.  10
    Closed Normal Subgroups.James H. Schmerl - 2001 - Mathematical Logic Quarterly 47 (4):489-492.
    Let ℳ be a countable, recursively saturated model of Peano Arithmetic, and let Aut be its automorphism group considered as a topological group with the pointwise stabilizers of finite sets being the basic open subgroups. Kaye proved that the closed normal subgroups are precisely the obvious ones, namely the stabilizers of invariant cuts. A proof of Kaye's theorem is given here which, although based on his proof, is different enough to yield consequences not obtainable from Kaye's proof.
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  17. Coinductive ℵ0-Categorical Theories.James H. Schmerl - 1990 - Journal of Symbolic Logic 55 (3):1130 - 1137.
  18. Decidability and ℵ0-Categoricity of Theories of Partially Ordered Sets.James H. Schmerl - 1980 - Journal of Symbolic Logic 45 (3):585 - 611.
    This paper is primarily concerned with ℵ 0 -categoricity of theories of partially ordered sets. It contains some general conjectures, a collection of known results and some new theorems on ℵ 0 -categoricity. Among the latter are the following. Corollary 3.3. For every countable ℵ 0 -categorical U there is a linear order of A such that $(\mathfrak{U}, is ℵ 0 -categorical. Corollary 6.7. Every ℵ 0 -categorical theory of a partially ordered set of finite width has a decidable theory. (...)
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  19.  9
    Generalizing Special Aronszajn Trees.James H. Schmerl - 1974 - Journal of Symbolic Logic 39 (4):732-740.
  20.  25
    Self-Embeddings of Computable Trees.Stephen Binns, Bjørn Kjos-Hanssen, Manuel Lerman, James H. Schmerl & Reed Solomon - 2008 - Notre Dame Journal of Formal Logic 49 (1):1-37.
    We divide the class of infinite computable trees into three types. For the first and second types, 0' computes a nontrivial self-embedding while for the third type 0'' computes a nontrivial self-embedding. These results are optimal and we obtain partial results concerning the complexity of nontrivial self-embeddings of infinite computable trees considered up to isomorphism. We show that every infinite computable tree must have either an infinite computable chain or an infinite Π01 antichain. This result is optimal and has connections (...)
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  21.  2
    Automorphism Groups of Saturated Models of Peano Arithmetic.Ermek S. Nurkhaidarov & James H. Schmerl - 2014 - Journal of Symbolic Logic 79 (2):561-584.
  22.  10
    Infinite Substructure Lattices of Models of Peano Arithmetic.James H. Schmerl - 2010 - Journal of Symbolic Logic 75 (4):1366-1382.
    Bounded lattices (that is lattices that are both lower bounded and upper bounded) form a large class of lattices that include all distributive lattices, many nondistributive finite lattices such as the pentagon lattice N₅, and all lattices in any variety generated by a finite bounded lattice. Extending a theorem of Paris for distributive lattices, we prove that if L is an ℵ₀-algebraic bounded lattice, then every countable nonstandard model of Peano Arithmetic has a cofinal elementary extension such that (...)
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  23.  17
    Theories with Recursive Models.Manuel Lerman & James H. Schmerl - 1979 - Journal of Symbolic Logic 44 (1):59-76.
  24.  40
    The Automorphism Group of a Resplendent Model.James H. Schmerl - 2012 - Archive for Mathematical Logic 51 (5-6):647-649.
  25.  6
    Uncountable Real Closed Fields with Pa Integer Parts.David Marker, James H. Schmerl & Charles Steinhorn - 2015 - Journal of Symbolic Logic 80 (2):490-502.
  26. Large Resplendent Models Generated by Indiscernibles.James H. Schmerl - 1989 - Journal of Symbolic Logic 54 (4):1382-1388.
  27.  3
    Automorphism Groups of Countable Arithmetically Saturated Models of Peano Arithmetic.James H. Schmerl - 2015 - Journal of Symbolic Logic 80 (4):1411-1434.
  28.  14
    An Elementary Sentence Which has Ordered Models.James H. Schmerl - 1972 - Journal of Symbolic Logic 37 (3):521-530.
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  29.  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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  30.  25
    Reverse Mathematics and Grundy Colorings of Graphs.James H. Schmerl - 2010 - Mathematical Logic Quarterly 56 (5):541-548.
    The relationship of Grundy and chromatic numbers of graphs in the context of Reverse Mathematics is investi-gated.
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  31.  9
    Coinductive $Aleph_0$-Categorical Theories.James H. Schmerl - 1990 - Journal of Symbolic Logic 55 (3):1130-1137.
  32.  12
    An Axiomatization for a Class of Two-Cardinal Models.James H. Schmerl - 1977 - Journal of Symbolic Logic 42 (2):174-178.
  33. CP‐Generic Expansions of Models of Peano Arithmetic.Athar Abdul-Quader & James H. Schmerl - 2022 - Mathematical Logic Quarterly 68 (2):171-177.
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  34. Undecidable Theories and Reverse Mathematics.James H. Schmerl - 2005 - In Stephen Simpson (ed.), Reverse Mathematics 2001. pp. 21--349.
  35.  7
    End Extensions of Models of Arithmetic.James H. Schmerl - 1992 - Notre Dame Journal of Formal Logic 33 (2):216-219.
  36.  45
    A Reflection Principle and its Applications to Nonstandard Models.James H. Schmerl - 1995 - Journal of Symbolic Logic 60 (4):1137-1152.
  37.  18
    Nondiversity in Substructures.James H. Schmerl - 2008 - Journal of Symbolic Logic 73 (1):193-211.
    For a model M of Peano Arithmetic, let Lt(M) be the lattice of its elementary substructures, and let Lt⁺(M) be the equivalenced lattice (Lt(M), ≅M), where ≅M is the equivalence relation of isomorphism on Lt(M). It is known that Lt⁺(M) is always a reasonable equivalenced lattice. Theorem. Let L be a finite distributive lattice and let (L,E) be reasonable. If M₀ is a nonstandard prime model of PA, then M₀ has a confinal extension M such that Lt⁺(M) ≅ (L,E). A (...)
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  38.  20
    Remarks on Self-Extending Models.James H. Schmerl - 1976 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 22 (1):509-512.
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  39.  20
    Theories Having Finitely Many Countable Homogeneous Models.James H. Schmerl - 1986 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 32 (7-9):131-131.
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  40.  20
    Partitioning Large Vector Spaces.James H. Schmerl - 2003 - Journal of Symbolic Logic 68 (4):1171-1180.
  41.  16
    A Weakly Definable Type Which is Not Definable.James H. Schmerl - 1993 - Archive for Mathematical Logic 32 (6):463-468.
    For each completion of Peano Arithmetic there is a weakly definable type which is not definable.
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  42.  8
    Deciding the Chromatic Numbers of Algebraic Hypergraphs.James H. Schmerl - 2018 - Journal of Symbolic Logic 83 (1):128-145.
    For each infinite cardinalκ, the set of algebraic hypergraphs having chromatic number no larger thanκis decidable.
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  43.  17
    A Generalization of Sierpiński's Paradoxical Decompositions: Coloring Semialgebraic Grids.James H. Schmerl - 2012 - Journal of Symbolic Logic 77 (4):1165-1183.
    A structure A = (A; E₀, E₁ , . . . , ${E_{n - 2}}$) is an n-grid if each E i is an equivalence relation on A and whenver X and Y are equivalence classes of, repectively, distinct E i and E j , then X ∩ Y is finite. A coloring χ : A → n is acceptable if whenver X is an equivalence class of E i , then {ϰ Є X: χ(ϰ) = i} is finite. If (...)
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  44.  19
    Partially Ordered Sets and the Independence Property.James H. Schmerl - 1989 - Journal of Symbolic Logic 54 (2):396-401.
    No theory of a partially ordered set of finite width has the independence property, generalizing Poizat's corresponding result for linearly ordered sets. In fact, a question of Poizat concerning linearly ordered sets is answered by showing, moreover, that no theory of a partially ordered set of finite width has the multi-order property. It then follows that a distributive lattice is not finite-dimensional $\operatorname{iff}$ its theory has the independence property $\operatorname{iff}$ its theory has the multi-order property.
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  45.  5
    Acceptable Colorings of Indexed Hyperspaces.James H. Schmerl - 2018 - Journal of Symbolic Logic 83 (4):1644-1666.
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  46.  13
    Binary Relational Structures Having Only Countably Many Nonisomorphic Substructures.Dugald Macpherson & James H. Schmerl - 1991 - Journal of Symbolic Logic 56 (3):876-884.
  47.  8
    Difference Sets and Recursion Theory.James H. Schmerl - 1998 - Mathematical Logic Quarterly 44 (4):515-521.
    There is a recursive set of natural numbers which is the difference set of some recursively enumerable set but which is not the difference set of any recursive set.
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  48.  4
    More Automorphism Groups of Countable, Arithmetically Saturated Models of Peano Arithmetic.James H. Schmerl - 2018 - Notre Dame Journal of Formal Logic 59 (4):491-496.
    There is an infinite set T of Turing-equivalent completions of Peano Arithmetic such that whenever M and N are nonisomorphic countable, arithmetically saturated models of PA and Th, Th∈T, then Aut≇Aut.
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  49.  30
    Decidability and Finite Axiomatizability of Theories of ℵ0-Categorical Partially Ordered Sets.James H. Schmerl - 1981 - Journal of Symbolic Logic 46 (1):101 - 120.
    Every ℵ 0 -categorical partially ordered set of finite width has a finitely axiomatizable theory. Every ℵ 0 -categorical partially ordered set of finite weak width has a decidable theory. This last statement constitutes a major portion of the complete (with three exceptions) characterization of those finite partially ordered sets for which any ℵ 0 -categorical partially ordered set not embedding one of them has a decidable theory.
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  50.  4
    Minimal Elementary End Extensions.James H. Schmerl - 2017 - Archive for Mathematical Logic 56 (5-6):541-553.
    Suppose that M⊧PA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal M}\models \mathsf{PA}$$\end{document} and X⊆P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak X} \subseteq {\mathcal P}$$\end{document}. If M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal M}$$\end{document} has a finitely generated elementary end extension N≻endM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal N}\succ _\mathsf{end} {\mathcal M}$$\end{document} such that {X∩M:X∈Def}=X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{X \cap M : X \in {{\mathrm{Def}}}\} = (...)
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