Switch to: References

Add citations

You must login to add citations.
  1. Automorphism Groups of Saturated Models of Peano Arithmetic.Ermek S. Nurkhaidarov & James H. Schmerl - 2014 - Journal of Symbolic Logic 79 (2):561-584.
    Letκbe the cardinality of some saturated model of Peano Arithmetic. There is a set of${2^{{\aleph _0}}}$saturated models of PA, each having cardinalityκ, such that wheneverMandNare two distinct models from this set, then Aut(${\cal M}$) ≇ Aut ($${\cal N}$$).
    Download  
     
    Export citation  
     
    Bookmark   2 citations  
  • 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 (...)
    Download  
     
    Export citation  
     
    Bookmark   2 citations  
  • (2 other versions)2005–06 Winter Meeting of the Association for Symbolic Logic.Valentina Harizanov - 2006 - Bulletin of Symbolic Logic 12 (4):613-624.
    Download  
     
    Export citation  
     
    Bookmark  
  • 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.
    Download  
     
    Export citation  
     
    Bookmark   5 citations  
  • Constant Regions in Models of Arithmetic.Tin Lok Wong - 2015 - Notre Dame Journal of Formal Logic 56 (4):603-624.
    This paper introduces a new theory of constant regions, which generalizes that of interstices, in nonstandard models of arithmetic. In particular, we show that two homogeneity notions introduced by Richard Kaye and the author, namely, constantness and pregenericity, are equivalent. This led to some new characterizations of generic cuts in terms of existential closedness.
    Download  
     
    Export citation  
     
    Bookmark  
  • (1 other version)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.
    Download  
     
    Export citation  
     
    Bookmark  
  • (1 other version)Automorphism Groups of Countable Arithmetically Saturated Models of Peano Arithmetic.James H. Schmerl - 2015 - Journal of Symbolic Logic 80 (4):1411-1434.
    If${\cal M},{\cal N}$are countable, arithmetically saturated models of Peano Arithmetic and${\rm{Aut}}\left( {\cal M} \right) \cong {\rm{Aut}}\left( {\cal N} \right)$, then the Turing-jumps of${\rm{Th}}\left( {\cal M} \right)$and${\rm{Th}}\left( {\cal N} \right)$are recursively equivalent.
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  • Interstitial and pseudo gaps in models of Peano Arithmetic.Ermek S. Nurkhaidarov - 2010 - Mathematical Logic Quarterly 56 (2):198-204.
    In this paper we study the automorphism groups of models of Peano Arithmetic. Kossak, Kotlarski, and Schmerl [9] shows that the stabilizer of an unbounded element a of a countable recursively saturated model of Peano Arithmetic M is a maximal subgroup of Aut if and only if the type of a is selective. We extend this result by showing that if M is a countable arithmetically saturated model of Peano Arithmetic, Ω ⊂ M is a very good interstice, and a (...)
    Download  
     
    Export citation  
     
    Bookmark   2 citations  
  • Closed Normal Subgroups of the Automorphism Group of a Saturated Model of Peano Arithmetic.Ermek S. Nurkhaidarov & Erez Shochat - 2016 - Notre Dame Journal of Formal Logic 57 (1):127-139.
    In this paper we discuss automorphism groups of saturated models and boundedly saturated models of $\mathsf{PA}$. We show that there are saturated models of $\mathsf{PA}$ of the same cardinality with nonisomorphic automorphism groups. We then show that every saturated model of $\mathsf{PA}$ has short saturated elementary cuts with nonisomorphic automorphism groups.
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  • Decoding in the automorphism group of a recursively saturated model of arithmetic.Ermek Nurkhaidarov - 2015 - Mathematical Logic Quarterly 61 (3):179-188.
    The main result of this paper partially answers a question raised in about the existence of countable just recursively saturated models of Peano Arithmetic with non‐isomorphic automorphism groups. We show the existence of infinitely many countable just recursively saturated models of Peano Arithmetic such that their automorphism groups are not topologically isomorphic. We also discuss maximal open subgroups of the automorphism group of a countable arithmetically saturated model of in a very good interstice.
    Download  
     
    Export citation  
     
    Bookmark  
  • (2 other versions)Of the association for symbolic logic.Valentina Harizanov - 2006 - Bulletin of Symbolic Logic 12 (4):613-624.
    Download  
     
    Export citation  
     
    Bookmark  
  • Automorphism Groups of Arithmetically Saturated Models.Ermek S. Nurkhaidarov - 2006 - Journal of Symbolic Logic 71 (1):203 - 216.
    In this paper we study the automorphism groups of countable arithmetically saturated models of Peano Arithmetic. The automorphism groups of such structures form a rich class of permutation groups. When studying the automorphism group of a model, one is interested to what extent a model is recoverable from its automorphism group. Kossak-Schmerl [12] show that ifMis a countable, arithmetically saturated model of Peano Arithmetic, then Aut(M) codes SSy(M). Using that result they prove:Let M1. M2be countable arithmetically saturated models of Peano (...)
    Download  
     
    Export citation  
     
    Bookmark   7 citations  
  • Strongly maximal subgroups determined by elements in interstices.Teresa Bigorajska - 2003 - Mathematical Logic Quarterly 49 (1):101-108.
    Continuing the earlier research in [1] and [4] we work out a class of interstices in countable arithmetically saturated models of PA in which selective types are realized and a class of interstices in which 2-indiscernible types are realized.
    Download  
     
    Export citation  
     
    Bookmark   2 citations