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A standard model of Peano arithmetic with no conservative elementary extension, preprint
, 2006
"... Abstract. The principal result of this paper answers a longstanding question in the model theory of arithmetic [KS, Question 7] by showing that there exists an uncountable arithmetically closed family A of subsets of the set ω of natural numbers such that the expansion ΩA: = (ω, +, ·, X)X∈A of the ..."
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Abstract. The principal result of this paper answers a longstanding question in the model theory of arithmetic [KS, Question 7] by showing that there exists an uncountable arithmetically closed family A of subsets of the set ω of natural numbers such that the expansion ΩA: = (ω, +, ·, X)X∈A of the standard model of Peano arithmetic has no conservative elementary extension, i.e., for any elementary extension Ω ∗ A = (ω∗, · · ·) of ΩA, there is a subset of ω ∗ that is parametrically definable in Ω ∗ A but whose intersection with ω is not a member of A. Inspired by a recent question of Gitman and Hamkins, we also show that the aforementioned family A can be arranged to further satisfy the curious property that forcing with the quotient Boolean algebra A/F IN (where F IN is the ideal of finite sets) collapses ℵ1 when viewed as a notion of forcing. 1.
Undefinable classes and definable elements in models of set theory and arithmetic
 Proc. Amer. Math. Soc
, 1988
"... ABSTRACT. Every countable model M of PA or ZFC, by a theorem of S. Simpson, has a "class " X which has the curious property: Every element of the expanded structure (M, X) is definable. Here we prove: THEOREM A. Every completion T of PA has a countable model M (indeed there are 2 " ma ..."
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ABSTRACT. Every countable model M of PA or ZFC, by a theorem of S. Simpson, has a "class " X which has the curious property: Every element of the expanded structure (M, X) is definable. Here we prove: THEOREM A. Every completion T of PA has a countable model M (indeed there are 2 " many such M 's for each T) which is not pointwise definable and yet becomes pointwise definable upon adjoining any undefinable class X to M. THEOREM B. Let M 1 = ZF + "V = HOD " be a wellfounded model of any cardinality. There exists an undefinable class X such that the definable points of M and (M, X) coincide. THEOREM C. Let M t = PA or ZF +"V = HOD". There exists an undefinable class X such that the definable points of M and (M, X) coincide if one of the conditions below is satisfied. (A) The definable elements o/M are cofinal in M. (B) M is recursively saturated and cf (M) = uj. Let M be a model of Peano arithmetic PA (or ZermeloFraenkel set theory ZF).
Automorphisms of models of arithmetic: a unified view
 Ann. Pure Appl. Logic
"... We develop the method of iterated ultrapower representation to provide a unified and perspicuous approach for building automorphisms of countable recursively saturated models of Peano arithmetic P A. In particular, we use this method to prove Theorem A below, which confirms a long standing conjectur ..."
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We develop the method of iterated ultrapower representation to provide a unified and perspicuous approach for building automorphisms of countable recursively saturated models of Peano arithmetic P A. In particular, we use this method to prove Theorem A below, which confirms a long standing conjecture of James Schmerl. Theorem A. If M is a countable recursively saturated model of P A in which N is a strong cut, then for any M0 ≺ M there is an automorphism j of M such that the fixed point set of j is isomorphic to M0. We also finetune a number of classical results. One of our typical results in this direction is Theorem B below, which generalizes a theorem of KayeKossakKotlarski (in what follows Aut(X) is the automorphism group of the structure X, and Q is the ordered set of rationals). Theorem B. Suppose M is a countable recursively saturated model of P A in which N is a strong cut. There is a group embedding j ↦ → ˆj from
The automorphism group of an arithmetically saturated model of Peano arithmetic
 J. London Math. Soc
, 1995
"... One of the main goals in the study of the automorphism group Aut {Jt) of a countable, recursively saturated model Jt of Peano Arithmetic is to determine to what extent (the isomorphism type of) Jt is recoverable from (the isomorphism type of) Aut(^). A countable, recursively saturated model Jt of PA ..."
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One of the main goals in the study of the automorphism group Aut {Jt) of a countable, recursively saturated model Jt of Peano Arithmetic is to determine to what extent (the isomorphism type of) Jt is recoverable from (the isomorphism type of) Aut(^). A countable, recursively saturated model Jt of PA is characterized up to isomorphism by two invariants: its firstorder theory Th(^) and its standard system SSy {Jt). At present, there seems to be no indication of how to recover any information about Th {Jt) from Aut {Jt) with the exception of whether or not Th {Jt) is True Arithmetic. We define the notion of arithmetically saturated in Definition 1.7; however, a model Jt of PA is arithmetically saturated if and only if it is recursively saturated and the standard cut is a strong cut. The following is our main theorem. THEOREM. Suppose that Jtx and Jt2 are countable, arithmetically saturated models of PA such that Aut(^) s Aut {Jt2). Then SSy {Jtx) = SSy {Jt2). In the Theorem, it suffices to assume that Jt2 is just recursively saturated. For, as shown by Lascar [8], if Jtx and Jt2 are countable, recursively saturated models of PA
THE COMPLEXITY OF CLASSIFICATION PROBLEMS FOR MODELS OF ARITHMETIC
, 908
"... Abstract. We observe that the classification problem for countable models of arithmetic is Borel complete. On the other hand, the classification problems for finitely generated models of arithmetic and for recursively saturated models of arithmetic are Borel; we investigate the precise complexity of ..."
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Abstract. We observe that the classification problem for countable models of arithmetic is Borel complete. On the other hand, the classification problems for finitely generated models of arithmetic and for recursively saturated models of arithmetic are Borel; we investigate the precise complexity of each of these. Finally, we show that the classification problem for pairs of recursively saturated models and for automorphisms of a fixed recursively saturated model are Borel complete. 1.
WEAKLY DEFINABLE TYPES
"... ABSTRACT. We study some generalizations of the notion of a definable type, first in an abstract setting in terms of ultrafilters on certain Boolean algebras, and then as applied to model theory. The notion of a weakly definable ultrafilter or type was developed by one of the authors [K] in a study o ..."
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ABSTRACT. We study some generalizations of the notion of a definable type, first in an abstract setting in terms of ultrafilters on certain Boolean algebras, and then as applied to model theory. The notion of a weakly definable ultrafilter or type was developed by one of the authors [K] in a study of models of arithmetic. It generalizes the notion of a definable type; and just as this latter notion has interesting properties in a much more general context, especially in stability theory, it seemed worthwhile to investigate weakly definable types in a general modeltheoretic setting. A goal of this paper is to present the results of our investigations on these lines. It is natural to ask why such notions turn up both in arithmetic and in elementary stability theory. Ressayre, for example, in a review [R] of Gaifman's paper [G], says...although the notion of definable type was introduced by Gaifman in the study of PA, which is the most unstable theory, this notion turned out to be a fundamental one for stable theories. And minimal as well as uniform types also correspond more or less to properties important in the stable case. I expect
CONSTRUCTING κLIKE MODELS OF ARITHMETIC
"... A model (M,�,…)isκlike if M has cardinality κ but, for all a�M, the cardinality of �x�M: x�a� is strictly less than κ. In this paper we shall give constructions of κlike models of arithmetic satisfying an arbitrarily large finite part of PA but not PA itself, for various singular cardinals κ. The ..."
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A model (M,�,…)isκlike if M has cardinality κ but, for all a�M, the cardinality of �x�M: x�a� is strictly less than κ. In this paper we shall give constructions of κlike models of arithmetic satisfying an arbitrarily large finite part of PA but not PA itself, for various singular cardinals κ. The main results are: (1) for each countable nonstandard M�Π � �Th(PA) with arbitrarily large initial segments satisfying PA and each uncountable κ of cofinality ω there is a cofinal extension K of M which is κlike; also hierarchical variants of this result for Π n �Th(PA); and (2) for every n�1, every singular κ and every M�BΣ n �exp��IΣ n there is a κlike model K elementarily equivalent to M.
Arithmetic and the Incompleteness Theorems
, 2000
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© Hindawi Publishing Corp. RIGID LEFT NOETHERIAN RINGS
, 2003
"... We prove that any rigid left Noetherian ring is either a domain or isomorphic to some ring Zpn of integers modulo a prime power pn. 2000 Mathematics Subject Classification: 16P40, 16W20, 16W25. Let R be an associative ring. A map σ: R → R is called a ring endomorphism if σ(x+y) = σ(x)+σ(y) and σ(xy ..."
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We prove that any rigid left Noetherian ring is either a domain or isomorphic to some ring Zpn of integers modulo a prime power pn. 2000 Mathematics Subject Classification: 16P40, 16W20, 16W25. Let R be an associative ring. A map σ: R → R is called a ring endomorphism if σ(x+y) = σ(x)+σ(y) and σ(xy) = σ(x)σ(y) for all elements a,b ∈ R. A ring R is said to be rigid if it has only the trivial ring endomorphisms, that is, identity idR and zero 0R. Rigid left Artinian rings were described by Maxson [9] and McLean [11]. Friger [4, 6] has constructed an example of a noncommutative rigid ring R with the additive group R+ of finite Prüfer rank. A characterization for rigid rings of finite rank was obtained by the author in [1]. Some aspects of a ring rigidity has been studied by Suppa [12, 13], Friger [5], and the author [2]. In this paper, we study rigid left Noetherian rings and prove the following theorem. Theorem 1. Let R be a left Noetherian ring. Then R is a rigid ring if and only if R Zpt (p is a prime, t ∈N) or it is a rigid domain. All rings are assumed to be associative and, as a rule, with an identity element. For a