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Richard Kaye [29]Richard W. Kaye [1]
  1.  47
    On Interpretations of Arithmetic and Set Theory.Richard Kaye & Tin Lok Wong - 2007 - Notre Dame Journal of Formal Logic 48 (4):497-510.
    This paper starts by investigating Ackermann's interpretation of finite set theory in the natural numbers. We give a formal version of this interpretation from Peano arithmetic (PA) to Zermelo-Fraenkel set theory with the infinity axiom negated (ZF−inf) and provide an inverse interpretation going the other way. In particular, we emphasize the precise axiomatization of our set theory that is required and point out the necessity of the axiom of transitive containment or (equivalently) the axiom scheme of ∈-induction. This clarifies the (...)
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  2.  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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  3.  9
    Diophantine Induction.Richard Kaye - 1990 - Annals of Pure and Applied Logic 46 (1):1-40.
    We show that Matijasevič's Theorem on the diophantine representation of r.e. predicates is provable in the subsystem I ∃ - 1 of Peano Arithmetic formed by restricting the induction scheme to diophantine formulas with no parameters. More specifically, I ∃ - 1 ⊢ IE - 1 + E ⊢ Matijasevič's Theorem where IE - 1 is the scheme of parameter-free bounded existential induction and E is an ∀∃ axiom expressing the existence of a function of exponential growth. We conclude by (...)
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  4.  32
    The Mathematics of Logic: A Guide to Completeness Theorems and Their Applications.Richard Kaye - 2007 - Cambridge University Press.
    This undergraduate textbook covers the key material for a typical first course in logic, in particular presenting a full mathematical account of the most important result in logic, the Completeness Theorem for first-order logic. Looking at a series of interesting systems, increasing in complexity, then proving and discussing the Completeness Theorem for each, the author ensures that the number of new concepts to be absorbed at each stage is manageable, whilst providing lively mathematical applications throughout. Unfamiliar terminology is kept to (...)
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  5.  9
    The Model Theory of Generic Cuts.Tin Lok Wong & Richard Kaye - 2015 - In Andrés Villaveces, Roman Kossak, Juha Kontinen & Åsa Hirvonen (eds.), The Model Theory of Generic Cuts. De Gruyter. pp. 281-296.
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  6.  94
    Truth in Generic Cuts.Richard Kaye & Tin Lok Wong - 2010 - Annals of Pure and Applied Logic 161 (8):987-1005.
    In an earlier paper the first author initiated the study of generic cuts of a model of Peano arithmetic relative to a notion of an indicator in the model. This paper extends that work. We generalise the idea of an indicator to a related neighbourhood system; this allows the theory to be extended to one that includes the case of elementary cuts. Most results transfer to this more general context, and in particular we obtain the idea of a generic cut (...)
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  7.  19
    Generic Cuts in Models of Arithmetic.Richard Kaye - 2008 - Mathematical Logic Quarterly 54 (2):129-144.
    We present some general results concerning the topological space of cuts of a countable model of arithmetic given by a particular indicator Y.The notion of “indicator” is de.ned in a novel way, without initially specifying what property is indicated and is used to de.ne a topological space of cuts of the model. Various familiar properties of cuts are investigated in this sense, and several results are given stating whether or not the set of cuts having the property is comeagre.A new (...)
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  8.  11
    Parameter‐Free Universal Induction.Richard Kaye - 1989 - Mathematical Logic Quarterly 35 (5):443-456.
    No categories
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  9.  34
    End-Extensions Preserving Power Set.Thomas Forster & Richard Kaye - 1991 - Journal of Symbolic Logic 56 (1):323-328.
  10.  54
    A Generalization of Specker's Theorem on Typical Ambiguity.Richard Kaye - 1991 - Journal of Symbolic Logic 56 (2):458-466.
    We generalize Specker's theorem on typical ambiguity, that NF and TST + Ambiguity have the same stratified consequences, to the subschemes Amb(Γ) of ambiguity restricted to classes of sentences Γ with certain natural closure conditions.
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  11.  26
    Parameter-Free Universal Induction.Richard Kaye - 1989 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 35 (5):443-456.
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  12.  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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  13.  24
    The Arithmetic of Cuts in Models of Arithmetic.Richard Kaye - 2013 - Mathematical Logic Quarterly 59 (4-5):332-351.
  14.  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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  15.  6
    On Cofinal Extensions of Models of Fragments of Arithmetic.Richard Kaye - 1991 - Notre Dame Journal of Formal Logic 32 (3):399-408.
  16.  16
    The Theory of $\kappa$ -Like Models of Arithmetic.Richard Kaye - 1995 - Notre Dame Journal of Formal Logic 36 (4):547-559.
    A model is said to be -like if but for all , . In this paper, we shall study sentences true in -like models of arithmetic, especially in the cases when is singular. In particular, we identify axiom schemes true in such models which are particularly `natural' from a combinatorial or model-theoretic point of view and investigate the properties of models of these schemes.
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  17.  26
    Normal Subgroups of Nonstandard Symmetric and Alternating Groups.John Allsup & Richard Kaye - 2007 - Archive for Mathematical Logic 46 (2):107-121.
    Let ${\mathfrak{M}}$ be a nonstandard model of Peano Arithmetic with domain M and let ${n \in M}$ be nonstandard. We study the symmetric and alternating groups S n and A n of permutations of the set ${\{0,1,\ldots,n-1\}}$ internal to ${\mathfrak{M}}$ , and classify all their normal subgroups, identifying many externally defined such normal subgroups in the process. We provide evidence that A n and S n are not split extensions by these normal subgroups, by showing that any such complement if (...)
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  18.  43
    Model-Theoretic Properties Characterizing Peano Arithmetic.Richard Kaye - 1991 - Journal of Symbolic Logic 56 (3):949-963.
    Let $\mathscr{L} = \{0, 1, +, \cdot, <\}$ be the usual first-order language of arithmetic. We show that Peano arithmetic is the least first-order L-theory containing IΔ0 + exp such that every complete extension T of it has a countable model K satisfying. (i) K has no proper elementary substructures, and (ii) whenever $L \prec K$ is a countable elementary extension there is $\bar{L} \prec L$ and $\bar{K} \subseteq_\mathrm{e} \bar{L}$ such that $K \prec_{\mathrm{cf}}\bar{K}$ . Other model-theoretic conditions similar to (i) (...)
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  19.  7
    Model-Theoretic Properties Characterizing Peano Arithmetic.Richard Kaye - 1991 - Journal of Symbolic Logic 56 (3):949-963.
    Let $\mathscr{L} = \{0, 1, +, \cdot, <\}$ be the usual first-order language of arithmetic. We show that Peano arithmetic is the least first-order $\mathscr{L}$-theory containing $I\Delta_0 + \exp$ such that every complete extension $T$ of it has a countable model $K$ satisfying. $K$ has no proper elementary substructures, and whenever $L \prec K$ is a countable elementary extension there is $\bar{L} \prec L$ and $\bar{K} \subseteq_\mathrm{e} \bar{L}$ such that $K \prec_{\mathrm{cf}}\bar{K}$. Other model-theoretic conditions similar to and are also discussed (...)
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  20.  8
    Infinitary Definitions of Equivalence Relations in Models of PA.Richard Kaye - 1997 - Annals of Pure and Applied Logic 89 (1):37-43.
  21.  13
    Book Review: T. E. Forster. Set Theory with a Universal Set: Exploring an Untyped Universe. [REVIEW]Richard Kaye - 1993 - Notre Dame Journal of Formal Logic 34 (2):302-309.
  22.  7
    Hilbert's Tenth Problem for Weak Theories of Arithmetic.Richard Kaye - 1993 - Annals of Pure and Applied Logic 61 (1-2):63-73.
    Hilbert's tenth problem for a theory T asks if there is an algorithm which decides for a given polynomial p() from [] whether p() has a root in some model of T. We examine some of the model-theoretic consequences that an affirmative answer would have in cases such as T = Open Induction and others, and apply these methods by providing a negative answer in the cases when T is some particular finite fragment of the weak theories IE1 or IU-1.
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  23.  89
    Amphi-ZF: Axioms for Conway Games.Michael Cox & Richard Kaye - 2012 - Archive for Mathematical Logic 51 (3-4):353-371.
    A theory of two-sided containers, denoted ZF2, is introduced. This theory is then shown to be synonymous to ZF in the sense of Visser (2006), via an interpretation involving Quine pairs. Several subtheories of ZF2, and their relationships with ZF, are also examined. We include a short discussion of permutation models (in the sense of Rieger–Bernays) over ZF2. We close with highlighting some areas for future research, mostly motivated by the need to understand non-wellfounded games.
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  24.  76
    Transplendent Models: Expansions Omitting a Type.Fredrik Engström & Richard W. Kaye - 2012 - Notre Dame Journal of Formal Logic 53 (3):413-428.
    We expand the notion of resplendency to theories of the kind T + p", where T is a fi rst-order theory and p" expresses that the type p is omitted. We investigate two di erent formulations and prove necessary and sucient conditions for countable recursively saturated models of PA. Some of the results in this paper can be found in one of the author's doctoral thesis [3].
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  25.  21
    A Diophantine Undecidable Subsystem of Arithmetic with No Induction Axioms.Richard Kaye - unknown
  26.  9
    Circularity in Soundness and Completeness.Richard Kaye - 2014 - Bulletin of Symbolic Logic 20 (1):24-38.
  27.  1
    Interpretations Between Ω-Logic and Second-Order Arithmetic.Richard Kaye - 2014 - Journal of Symbolic Logic 79 (3):845-858.
    This paper addresses the structures and ), whereMis a nonstandard model of PA andωis the standard cut. It is known that ) is interpretable in. Our main technical result is that there is an reverse interpretation of in ) which is ‘local’ in the sense of Visser [11]. We also relate the model theory of to the study of transplendent models of PA [2].This yields a number of model theoretic results concerning theω-models and their standard systems SSy, including the following.•$\left (...)
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  28.  15
    Petr Hájek and Pavel Pudlák. Metamathematics of First-Order Arithmetic. Perspectives in Mathematical Logic. Springer-Verlag, Berlin Etc. 1993, Xiv + 460 Pp. [REVIEW]Richard Kaye - 1995 - Journal of Symbolic Logic 60 (4):1317-1320.