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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


by Simone Bonnafous vii

by Patrick C´egielski 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 n-Induction
by A. Cord´on–Franco, A. Fern´andez–Margarit, and F.F. Lara–Mart´ın 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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