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Studies in Weak Arithmetics

Patrick Cégielski

The field of Weak arithmetics is application of logical methods to Number Theory, developed by mathematicians, philosophers, and Theoretical Computer Scientists. In this volume, after a general presentation of weak arithmetics, the following topics are studied: the properties of integers of a real closed field equipped with exponentiation; conservation results for the induction schema restricted to first-order formulas with a finite number of alternations of quantifiers; a survey on a class of tools, called pebble games, used in finite model theory; the fact that reals e and π have approximations expressed by first-order formulas using bounded quantifiers; properties on infinite pictures depending on the universe of sets used; a language that simulates in a sufficiently nice manner all algorithms of a certain restricted class; the logical complexity of the axiom of infinity in some variants of set theory without the axiom of foundation; and the complexity to determine whether a trace is included in another one.

Patrick Cégielski is professor at Université Paris-Est Créteil IUT de Sénart-Fontainebleau

December 2009

Contents

  • Foreword by Simone Bonnafous vii
  • Introduction by Patrick Cégielski 1
  • 1 On the Arithmetization of Real Fields with Exponentiation by Sedki Boughattas and Jean-Pierre Ressayre 13
  • 2 On Conservation Results for Parameter-Free Πn-Induction by A. Cordón-Franco, A. Fernández-Margarit, and F.F. Lara-Martín 49
  • 3 Pebble Games for Logics with Counting and Rank by Anuj Dawar and Bjarki Holm 99
  • 4 Rudimentary of Two Famous Real Numbers by Henri-Alex Esbelin 121
  • 5 Decision Problems for Recognizable Languages of Infinite Pictures by Olivier Finkel 127
  • 6 A Total Functional Programming Language Computing APRA by David Michel and Pierre Valarcher 153
  • 7 Statements of Ill-Founded Infinity in Set Theory by Eugenio Omodeo, Alberto Policriti, and Alexandru Tomescu 173
  • 8 On some Matching Problems in Trace Monoids by Karine Shahbazyan and Yuri Shoukourian 201

ISBN (Paperback): 9781575866024
ISBN (electronic): 9781575866826

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