16 found
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  1.  15
    Lou van den Dries. Tame Topology and o-Minimal Structures. London Mathematical Society Lecture Note Series, No. 248. Cambridge University Press, Cambridge, New York, and Oakleigh, Victoria, 1998, X + 180 Pp. [REVIEW]Alessandro Berarducci - 2000 - Bulletin of Symbolic Logic 6 (2):216-218.
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  2.  11
    A Descending Chain Condition for Groups Definable in o -Minimal Structures.Alessandro Berarducci, Margarita Otero, Yaa’cov Peterzil & Anand Pillay - 2005 - Annals of Pure and Applied Logic 134 (2):303-313.
    We prove that if G is a group definable in a saturated o-minimal structure, then G has no infinite descending chain of type-definable subgroups of bounded index. Equivalently, G has a smallest type-definable subgroup G00 of bounded index and G/G00 equipped with the “logic topology” is a compact Lie group. These results give partial answers to some conjectures of the fourth author.
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  3.  27
    The Interpretability Logic of Peano Arithmetic.Alessandro Berarducci - 1990 - Journal of Symbolic Logic 55 (3):1059-1089.
    PA is Peano arithmetic. The formula $\operatorname{Interp}_\mathrm{PA}(\alpha, \beta)$ is a formalization of the assertion that the theory PA + α interprets the theory PA + β (the variables α and β are intended to range over codes of sentences of PA). We extend Solovay's modal analysis of the formalized provability predicate of PA, Pr PA (x), to the case of the formalized interpretability relation $\operatorname{Interp}_\mathrm{PA}(x, y)$ . The relevant modal logic, in addition to the usual provability operator `□', has a (...)
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  4.  3
    Combinatorial Principles in Elementary Number Theory.Alessandro Berarducci & Benedetto Intrigila - 1991 - Annals of Pure and Applied Logic 55 (1):35-50.
    We prove that the theory IΔ0, extended by a weak version of the Δ0-Pigeonhole Principle, proves that every integer is the sum of four squares (Lagrange's theorem). Since the required weak version is derivable from the theory IΔ0 + ∀x (xlog(x) exists), our results give a positive answer to a question of Macintyre (1986). In the rest of the paper we consider the number-theoretical consequences of a new combinatorial principle, the ‘Δ0-Equipartition Principle’ (Δ0EQ). In particular we give a new proof, (...)
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  5.  3
    Intersection Theory for o-Minimal Manifolds.Alessandro Berarducci & Margarita Otero - 2001 - Annals of Pure and Applied Logic 107 (1-3):87-119.
    We develop an intersection theory for definable Cp-manifolds in an o-minimal expansion of a real closed field and we prove the invariance of the intersection numbers under definable Cp-homotopies . In particular we define the intersection number of two definable submanifolds of complementary dimensions, the Brouwer degree and the winding numbers. We illustrate the theory by deriving in the o-minimal context the Brouwer fixed point theorem, the Jordan-Brouwer separation theorem and the invariance of the Lefschetz numbers under definable Cp-homotopies. A. (...)
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  6.  10
    On the Provability Logic of Bounded Arithmetic.Rineke Verbrugge & Alessandro Berarducci - 1991 - Annals of Pure and Applied Logic 61 (1-2):75-93.
    Let PLω be the provability logic of IΔ0 + ω1. We prove some containments of the form L ⊆ PLω < Th(C) where L is the provability logic of PA and Th(C) is a suitable class of Kripke frames.
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  7.  19
    O-Minimal Spectra, Infinitesimal Subgroups and Cohomology.Alessandro Berarducci - 2007 - Journal of Symbolic Logic 72 (4):1177 - 1193.
    By recent work on some conjectures of Pillay, each definably compact group G in a saturated o-minimal expansion of an ordered field has a normal "infinitesimal subgroup" G00 such that the quotient G/G00, equipped with the "logic topology", is a compact (real) Lie group. Our first result is that the functor G → G/G00 sends exact sequences of definably compact groups into exact sequences of Lie groups. We then study the connections between the Lie group G/G00 and the o-minimal spectrum (...)
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  8.  25
    Transfer Methods for o-Minimal Topology.Alessandro Berarducci & Margarita Otero - 2003 - Journal of Symbolic Logic 68 (3):785-794.
    Let M be an o-minimal expansion of an ordered field. Let φ be a formula in the language of ordered domains. In this note we establish some topological properties which are transferred from $\varphi^M$ to $\varphi^R$ and vice versa. Then, we apply these transfer results to give a new proof of a result of M. Edmundo-based on the work of A. Strzebonski-showing the existence of torsion points in any definably compact group defined in an o-minimal expansion of an ordered field.
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  9.  48
    Corrigendum To: "Transfer Methods for O-Minimal Topology".Alessandro Berarducci, Mário Edmundo & Margarita Otero - 2007 - Journal of Symbolic Logic 72 (3):1079 - 1080.
  10.  15
    A Recursive Nonstandard Model of Normal Open Induction.Alessandro Berarducci & Margarita Otero - 1996 - Journal of Symbolic Logic 61 (4):1228-1241.
    Models of normal open induction are those normal discretely ordered rings whose nonnegative part satisfy Peano's axioms for open formulas in the language of ordered semirings. (Where normal means integrally closed in its fraction field.) In 1964 Shepherdson gave a recursive nonstandard model of open induction. His model is not normal and does not have any infinite prime elements. In this paper we present a recursive nonstandard model of normal open induction with an unbounded set of infinite prime elements.
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  11.  11
    O-Minimal Cohomology: Finiteness and Invariance Results.Alessandro Berarducci & Antongiulio Fornasiero - 2009 - Journal of Mathematical Logic 9 (2):167-182.
  12.  7
    An Effective Version of Wilkie's Theorem of the Complement and Some Effective o-Minimality Results.Alessandro Berarducci & Tamara Servi - 2004 - Annals of Pure and Applied Logic 125 (1-3):43-74.
    Wilkie 5 397) proved a “theorem of the complement” which implies that in order to establish the o-minimality of an expansion of with C∞ functions it suffices to obtain uniform bounds on the number of connected components of quantifier free definable sets. He deduced that any expansion of with a family of Pfaffian functions is o-minimal. We prove an effective version of Wilkie's theorem of the complement, so in particular given an expansion of the ordered field with finitely many C∞ (...)
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  13.  2
    Δ0-Complexity of the Relation y = Πi ⩽ nF.Alessandro Berarducci & Paola D'Aquino - 1995 - Annals of Pure and Applied Logic 75 (1-2):49-56.
    We prove that if G is a Δ 0 -definable function on the natural numbers and F = Π i = 0 n G , then F is also Δ 0 -definable. Moreover, the inductive properties of F can be proved inside the theory IΔ 0.
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  14.  35
    Cohomology of Groups in o-Minimal Structures: Acyclicity of the Infinitesimal Subgroup.Alessandro Berarducci - 2009 - Journal of Symbolic Logic 74 (3):891-900.
    By recent work on some conjectures of Pillay, each definably compact group in a saturated o-minimal structure is an extension of a compact Lie group by a torsion free normal divisible subgroup, called its infinitesimal subgroup. We show that the infinitesimal subgroup is cohomologically acyclic. This implies that the functorial correspondence between definably compact groups and Lie groups preserves the cohomology.
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  15.  8
    < I> Δ< Sub> 0-Complexity of the Relation< I> Y_=< I> Π_< Sub> I⩽ N< I> F_(< I> I).Alessandro Berarducci & Paola D'Aquino - 1995 - Annals of Pure and Applied Logic 75 (1):49-56.
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  16.  7
    Review: Lou van den Dries, Tame Topology and O-Minimal Structures. [REVIEW]Alessandro Berarducci - 2000 - Bulletin of Symbolic Logic 6 (2):216-218.