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Effective model theory: the number of models and their complexity
 MODELS AND COMPUTABILITY
, 1999
"... Effective model theory studies model theoretic notions with an eye towards issues of computability and effectiveness. We consider two possible starting points. If the basic objects are taken to be theories, then the appropriate effective version investigates decidable theories (the set of theorems i ..."
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Cited by 18 (6 self)
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Effective model theory studies model theoretic notions with an eye towards issues of computability and effectiveness. We consider two possible starting points. If the basic objects are taken to be theories, then the appropriate effective version investigates decidable theories (the set of theorems is computable) and decidable structures (ones with decidable theories). If the objects of initial interest are typical mathematical structures, then the starting point is computable structures. We present an introduction to both of these aspects of effective model theory organized roughly around the themes of the number and types of models of theories with particular attention to categoricity (as either a hypothesis or a conclusion) and the analysis of various computability issues in families of models.
Tennenbaum’s Theorem for Models of Arithmetic
, 2006
"... This paper discusses Tennenbaum’s Theorem in its original context of models of arithmetic, which states that there are no recursive nonstandard models of Peano Arithmetic. We focus on three separate areas: the historical background to the theorem; an understanding of the theorem and its relationship ..."
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Cited by 3 (0 self)
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This paper discusses Tennenbaum’s Theorem in its original context of models of arithmetic, which states that there are no recursive nonstandard models of Peano Arithmetic. We focus on three separate areas: the historical background to the theorem; an understanding of the theorem and its relationship with the Gödel–Rosser Theorem; and extensions of Tennenbaum’s theorem to diophantine problems in models of arthmetic, especially problems concerning which diophantine equations have roots in some model of a given theory of arithmetic.
Soare, Bounding homogeneous models
"... A Turing degree d is homogeneous bounding if every complete decidable (CD) theory has a ddecidable homogeneous model A, i.e., the elementary diagram D e (A) has degree d. It follows from results of Macintyre and Marker that every PA degree (i.e., every degree of a complete extension of Peano Arithm ..."
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Cited by 2 (1 self)
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A Turing degree d is homogeneous bounding if every complete decidable (CD) theory has a ddecidable homogeneous model A, i.e., the elementary diagram D e (A) has degree d. It follows from results of Macintyre and Marker that every PA degree (i.e., every degree of a complete extension of Peano Arithmetic) is homogeneous bounding. We prove that in fact a degree is homogeneous bounding if and only if it is a PA degree. We do this by showing that there is a single CD theory T such that every homogeneous model of T has a PA degree. 1
Models and recursivity
, 2002
"... It is commonly held that the natural numbers sequence 0, 1, 2,... possesses a unique structure. Yet by a well known model theoretic argument, there exist nonstandard models of the formal theory which is generally taken to axiomatize all of our practices and intentions pertaining to use of the term ..."
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It is commonly held that the natural numbers sequence 0, 1, 2,... possesses a unique structure. Yet by a well known model theoretic argument, there exist nonstandard models of the formal theory which is generally taken to axiomatize all of our practices and intentions pertaining to use of the term “natural number. ” Despite the structural similarity of this argument to the influential set theoretic indeterminacy argument based on the downward LöwenheimSkolem theorem, most theorists agree that the number theoretic version does not have skeptical consequences about the reference of “natural number ” analogous to the ‘relativity ’ Skolem claimed pertains to notions such as “uncountable ” and “cardinal. ” In this paper I argue that recent proposals by Shapiro, Lavine, McGee and Field which aim to distinguish the number and set theoretic indeterminacy arguments by locating extramathematical constraints on the interpretation of our number theoretic vocabulary are inadequate. I then suggest that if we
Arithmetic and the Incompleteness Theorems
, 2000
"... this paper please consult me first, via my home page. ..."
Real closures of models of weak arithmetic
, 2011
"... D’Aquino et al. (J. Symb. Log. 75(1)(2010)) have recently shown that every realclosed field with an integer part satisfying the arithmetic theory IΣ4 is recursively saturated, and that this theorem fails if IΣ4 is replaced by I∆0. We prove that the theorem holds if IΣ4 is replaced by weak subtheori ..."
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D’Aquino et al. (J. Symb. Log. 75(1)(2010)) have recently shown that every realclosed field with an integer part satisfying the arithmetic theory IΣ4 is recursively saturated, and that this theorem fails if IΣ4 is replaced by I∆0. We prove that the theorem holds if IΣ4 is replaced by weak subtheories of Buss ’ bounded arithmetic: PV or Σ b 1IND xk. It also holds for I∆0 (and even its subtheory IE 2) under a rather mild assumption on cofinality. On the other hand, it fails for the extension of IOpen by an axiom expressing the Bézout property, even under the same assumption on cofinality. A discretely ordered subring A of a realclosed field (henceforth often: rcf) R is an integer part of R if for every r ∈ R there exists a ∈ A such that a ≤ r < a + 1. It is wellknown that every rcf has an integer part [MR93], which is then a model of the weak arithmetic theory IOpen (induction for quantifierfree formulas in the language of ordered rings). On the other hand, every model of IOpen is an integer part of its real closure (or, more precisely, the real closure of its fraction field). Recently, d’Aquino et al. [DKS10] studied the question which rcfs have integer parts satisfying more arithmetic, e.g. Peano Arithmetic. It turns out
Czech Republic Real closures of models of weak arithmetic
, 2011
"... D’Aquino et al. (J. Symb. Log. 75(1)(2010)) have recently shown that every realclosed field with an integer part satisfying the arithmetic theory IΣ4 is recursively saturated, and that this theorem fails if IΣ4 is replaced by I∆0. We prove that the theorem holds if IΣ4 is replaced by weak subtheori ..."
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D’Aquino et al. (J. Symb. Log. 75(1)(2010)) have recently shown that every realclosed field with an integer part satisfying the arithmetic theory IΣ4 is recursively saturated, and that this theorem fails if IΣ4 is replaced by I∆0. We prove that the theorem holds if IΣ4 is replaced by weak subtheories of Buss ’ bounded arithmetic: PV or Σ b 1IND xk. It also holds for I∆0 (and even its subtheory IE 2) under a rather mild assumption on cofinality. On the other hand, it fails for the extension of IOpen by an axiom expressing the Bézout property, even under the same assumption on cofinality. A discretely ordered subring A of a realclosed field (henceforth often: rcf) R is an integer part of R if for every r ∈ R there exists a ∈ A such that a ≤ r < a + 1. It is wellknown that every rcf has an integer part [MR93], which is then a model of the weak arithmetic theory IOpen (induction for quantifierfree formulas in the language of ordered rings). On the other hand, every model of IOpen is an integer part of its real closure (or, more precisely, the real closure of its fraction field). Recently, d’Aquino et al. [DKS10] studied the question which rcfs have integer parts satisfying more arithmetic, e.g. Peano Arithmetic. It turns out
Volume 2X2. Number 2. April 19X4 DEGREES OF RECURSIVELY SATURATED MODELS BY
"... Abstract. Using relativizations of results of Goncharov and Peretyat'kin on decidable homogeneous models, we prove that if M is Ssaturated for some Scott set S, and F is an enumeration of S, then M has a presentation recursive in F. Applying this result we are able to classify degrees coding (i) th ..."
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Abstract. Using relativizations of results of Goncharov and Peretyat'kin on decidable homogeneous models, we prove that if M is Ssaturated for some Scott set S, and F is an enumeration of S, then M has a presentation recursive in F. Applying this result we are able to classify degrees coding (i) the reducts of models of PA to addition or multiplication, (ii) internally finite initial segments and (iii) nonstandard residue fields. We also use our results to simplify Solovay's characterization of degrees coding nonstandard models of Th(N). 0. Introduction. The classic theorem of Tennenbaum [T] says that there is no recursive nonstandard model of Peano arithmetic. Indeed, if (u, ffi, 0) N P, and (to, ffi, O) is nonstandard, then neither © nor O is recursive. In the above situation, (w, ffi) and («, O) are recursively saturated, and nowadays one formulates the theorem as the statement that there are no recursive, recursively saturated models of