49 found
Sort by:
  1. James H. Schmerl (forthcoming). Subsets Coded in Elementary End Extensions. Archive for Mathematical Logic.
    No categories
    Direct download (2 more)  
     
    My bibliography  
     
    Export citation  
  2. James H. Schmerl (2014). Cofinal Elementary Extensions. Mathematical Logic Quarterly 60 (1-2):12-20.
    No categories
    Direct download (2 more)  
     
    My bibliography  
     
    Export citation  
  3. Roman Kossak & James H. Schmerl (2012). On Cofinal Submodels and Elementary Interstices. 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 (...)
    Direct download (4 more)  
     
    My bibliography  
     
    Export citation  
  4. James H. Schmerl (2012). A Generalization of Sierpiński's Paradoxical Decompositions: Coloring Semialgebraic Grids. 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 (...)
    Direct download (4 more)  
     
    My bibliography  
     
    Export citation  
  5. James H. Schmerl (2012). Elementary Cuts in Saturated Models of Peano Arithmetic. 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} (...)
    Direct download (4 more)  
     
    My bibliography  
     
    Export citation  
  6. James H. Schmerl (2012). The Automorphism Group of a Resplendent Model. Archive for Mathematical Logic 51 (5-6):647-649.
    No categories
    Direct download (5 more)  
     
    My bibliography  
     
    Export citation  
  7. James H. Schmerl (2010). Infinite Substructure Lattices of Models of Peano Arithmetic. 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 (...)
    Direct download (4 more)  
     
    My bibliography  
     
    Export citation  
  8. James H. Schmerl (2010). Reverse Mathematics and Grundy Colorings of Graphs. Mathematical Logic Quarterly 56 (5):541-548.
    No categories
    Direct download (5 more)  
     
    My bibliography  
     
    Export citation  
  9. Stephen Binns, Bjørn Kjos-Hanssen, Manuel Lerman, James H. Schmerl & Reed Solomon (2008). Self-Embeddings of Computable Trees. 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 (...)
    Direct download (3 more)  
     
    My bibliography  
     
    Export citation  
  10. James H. Schmerl (2008). Nondiversity in Substructures. 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 (...)
    Direct download (4 more)  
     
    My bibliography  
     
    Export citation  
  11. James H. Schmerl (2005). Undecidable Theories and Reverse Mathematics. In Stephen Simpson (ed.), Reverse Mathematics 2001. 21--349.
    No categories
     
    My bibliography  
     
    Export citation  
  12. James H. Schmerl (2003). Partitioning Large Vector Spaces. Journal of Symbolic Logic 68 (4):1171-1180.
    Direct download (7 more)  
     
    My bibliography  
     
    Export citation  
  13. James H. Schmerl (2002). Automorphism Groups of Models of Peano Arithmetic. Journal of Symbolic Logic 67 (4):1249-1264.
    Direct download (6 more)  
     
    My bibliography  
     
    Export citation  
  14. James H. Schmerl (2002). Moving Intersticial Gaps. Mathematical Logic Quarterly 48 (2):283-296.
    No categories
    Direct download (4 more)  
     
    My bibliography  
     
    Export citation  
  15. James H. Schmerl (2002). Some Highly Saturated Models of Peano Arithmetic. Journal of Symbolic Logic 67 (4):1265-1273.
    Direct download (6 more)  
     
    My bibliography  
     
    Export citation  
  16. James H. Schmerl (2001). Closed Normal Subgroups. Mathematical Logic Quarterly 47 (4):489-492.
    No categories
    Direct download (4 more)  
     
    My bibliography  
     
    Export citation  
  17. James H. Schmerl (2000). Elementary Extensions of Models of Set Theory. Archive for Mathematical Logic 39 (7):509-514.
    A theorem of Enayat's concerning models of ZFC which had been proved using several different additional set-theoretical hypotheses is shown here to be absolute.
    No categories
    Direct download (4 more)  
     
    My bibliography  
     
    Export citation  
  18. James H. Schmerl (1998). Recursive Models and the Divisibility Poset. Notre Dame Journal of Formal Logic 39 (1):140-148.
    Direct download (5 more)  
     
    My bibliography  
     
    Export citation  
  19. James H. Schmerl (1998). Difference Sets and Recursion Theory. Mathematical Logic Quarterly 44 (4):515-521.
    No categories
    Direct download (3 more)  
     
    My bibliography  
     
    Export citation  
  20. James H. Schmerl (1998). What's the Difference? Annals of Pure and Applied Logic 93 (1-3):255-261.
    Direct download (3 more)  
     
    My bibliography  
     
    Export citation  
  21. Roman Kossak & James H. Schmerl (1995). Arithmetically Saturated Models of Arithmetic. 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.
    Direct download (4 more)  
     
    My bibliography  
     
    Export citation  
  22. James H. Schmerl (1995). PA( Aa ). Notre Dame Journal of Formal Logic 36 (4):560-569.
    The theory PA(aa), which is Peano Arithmetic in the context of stationary logic, is shown to be consistent. Moreover, the first-order theory of the class of finitely determinate models of PA(aa) is characterized.
    Direct download (5 more)  
     
    My bibliography  
     
    Export citation  
  23. James H. Schmerl (1995). A Reflection Principle and its Applications to Nonstandard Models. Journal of Symbolic Logic 60 (4):1137-1152.
  24. James H. Schmerl (1995). The Isomorphism Property for Nonstandard Universes. Journal of Symbolic Logic 60 (2):512-516.
    Direct download (6 more)  
     
    My bibliography  
     
    Export citation  
  25. Roman Kossak, Henryk Kotlarski & James H. Schmerl (1993). On Maximal Subgroups of the Automorphism Group of a Countable Recursively Saturated Model of PA. Annals of Pure and Applied Logic 65 (2):125-148.
    Direct download (3 more)  
     
    My bibliography  
     
    Export citation  
  26. James H. Schmerl (1993). A Weakly Definable Type Which is Not Definable. Archive for Mathematical Logic 32 (6):463-468.
    For each completion of Peano Arithmetic there is a weakly definable type which is not definable.
    No categories
    Direct download (4 more)  
     
    My bibliography  
     
    Export citation  
  27. James H. Schmerl (1992). End Extensions of Models of Arithmetic. Notre Dame Journal of Formal Logic 33 (2):216-219.
    Direct download (3 more)  
     
    My bibliography  
     
    Export citation  
  28. H. Jerome Keisler & James H. Schmerl (1991). Making the Hyperreal Line Both Saturated and Complete. 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 (...)
    Direct download (6 more)  
     
    My bibliography  
     
    Export citation  
  29. Roman Kossak & James H. Schmerl (1991). Minimal Satisfaction Classes with an Application to Rigid Models of Peano Arithmetic. Notre Dame Journal of Formal Logic 32 (3):392-398.
    Direct download (4 more)  
     
    My bibliography  
     
    Export citation  
  30. Dugald MacPherson & James H. Schmerl (1991). Binary Relational Structures Having Only Countably Many Nonisomorphic Substructures. Journal of Symbolic Logic 56 (3):876-884.
    Direct download (6 more)  
     
    My bibliography  
     
    Export citation  
  31. James H. Schmerl (1990). Coinductive $Aleph_0$-Categorical Theories. Journal of Symbolic Logic 55 (3):1130-1137.
    Direct download (3 more)  
     
    My bibliography  
     
    Export citation  
  32. James H. Schmerl (1990). Coinductive ℵ0-Categorical Theories. Journal of Symbolic Logic 55 (3):1130 - 1137.
    Direct download (3 more)  
     
    My bibliography  
     
    Export citation  
  33. James H. Schmerl (1989). Large Resplendent Models Generated by Indiscernibles. Journal of Symbolic Logic 54 (4):1382-1388.
    Direct download (6 more)  
     
    My bibliography  
     
    Export citation  
  34. James H. Schmerl (1989). Partially Ordered Sets and the Independence Property. 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.
    Direct download (6 more)  
     
    My bibliography  
     
    Export citation  
  35. Matt Kaufmann & James H. Schmerl (1987). Remarks on Weak Notions of Saturation in Models of Peano Arithmetic. Journal of Symbolic Logic 52 (1):129-148.
    Direct download (6 more)  
     
    My bibliography  
     
    Export citation  
  36. James H. Schmerl (1986). Theories Having Finitely Many Countable Homogeneous Models. Mathematical Logic Quarterly 32 (7‐9):131-131.
    No categories
    Direct download (3 more)  
     
    My bibliography  
     
    Export citation  
  37. James H. Schmerl (1985). Recursively Saturated Models Generated by Indiscernibles. Notre Dame Journal of Formal Logic 26 (2):99-105.
    Direct download (4 more)  
     
    My bibliography  
     
    Export citation  
  38. Matt Kaufmann & James H. Schmerl (1984). Saturation and Simple Extensions of Models of Peano Arithmetic. Annals of Pure and Applied Logic 27 (2):109-136.
    Direct download (3 more)  
     
    My bibliography  
     
    Export citation  
  39. James H. Schmerl & Stephen G. Simpson (1982). On the Role of Ramsey Quantifiers in First Order Arithmetic. Journal of Symbolic Logic 47 (2):423-435.
  40. James H. Schmerl (1981). Decidability and Finite Axiomatizability of Theories of ℵ0-Categorical Partially Ordered Sets. 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.
    Direct download (5 more)  
     
    My bibliography  
     
    Export citation  
  41. James H. Schmerl (1980). Decidability and ℵ0-Categoricity of Theories of Partially Ordered Sets. 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. (...)
    Direct download (5 more)  
     
    My bibliography  
     
    Export citation  
  42. Manuel Lerman & James H. Schmerl (1979). Theories with Recursive Models. Journal of Symbolic Logic 44 (1):59-76.
    Direct download (6 more)  
     
    My bibliography  
     
    Export citation  
  43. James H. Schmerl (1978). Extending Models of Arithmetic. Annals of Mathematical Logic 14 (2):89-109.
    No categories
    Direct download (2 more)  
     
    My bibliography  
     
    Export citation  
  44. James H. Schmerl (1977). An Axiomatization for a Class of Two-Cardinal Models. Journal of Symbolic Logic 42 (2):174-178.
    Direct download (6 more)  
     
    My bibliography  
     
    Export citation  
  45. James H. Schmerl (1976). On Κ-Like Structures Which Embed Stationary and Closed Unbounded Subsets. Annals of Mathematical Logic 10 (3-4):289-314.
    No categories
    Direct download (2 more)  
     
    My bibliography  
     
    Export citation  
  46. James H. Schmerl (1976). Remarks on Self‐Extending Models. Mathematical Logic Quarterly 22 (1):509-512.
    No categories
    Direct download (3 more)  
     
    My bibliography  
     
    Export citation  
  47. James H. Schmerl (1974). Generalizing Special Aronszajn Trees. Journal of Symbolic Logic 39 (4):732-740.
    Direct download (5 more)  
     
    My bibliography  
     
    Export citation  
  48. James H. Schmerl (1972). An Elementary Sentence Which has Ordered Models. Journal of Symbolic Logic 37 (3):521-530.
    Direct download (6 more)  
     
    My bibliography  
     
    Export citation  
  49. James H. Schmerl & Saharon Shelah (1972). On Power-Like Models for Hyperinaccessible Cardinals. Journal of Symbolic Logic 37 (3):531-537.
    Direct download (6 more)  
     
    My bibliography  
     
    Export citation