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New Studies in Weak Arithmetics

Edited by Patrick Cégielski, Charalampos Cornaros, and Costas Dimitracopoulos

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, Professor at University Paris-Est Créteil – IUT de Sénart Fontainebleau, is the coordinator of a European consortium of teams working on weak arithmetics, mainly from Armenia, Belgium, Czech Republic, Federation of Russia, France, United Kingdom, Greece, Israel, Italy, Poland, Portugal, Slovakia, Spain, Tunisia, and Ukraine.

Jointly published with Presses Universitaires du Pôle de Recherche et d'enseignement supérieur Paris-Est. See also Studies in Weak Arithmetics.

Table of Contents

  • Foreword by Luc Hittinger
  • Greetings by Petros Parianos
  • Introduction by Patrick Cégielski, Charalampos Cornaros, and Constantin Dimitracopoulos
  • I. Contributions
    1. A few questions concerning consistency and conservativeness, by Zofia Adamowicz
    2. In memoriam of Alan Robert Woods by Patrick Cégielski and Denis Richard
    3. Degres Ludiques : une introduction by C. Chalons and Jean-Pierre Ressayre
    4. Primes in Models of I0 + 1: Density in Henselizations by Paola D'Aquino and Angus Macintyre
    5. A New Proof of Tanaka's Theorem by Ali Enayat
    6. 0-definability of the denumerant with one plus three variables by Henri-Alex Esbelin
    7. Techniques in weak analysis for conservation results by Antonio M. Fernandes, Fernando Ferreira, and Gilda Ferreira
    8. Adding standardness to nonstandard arithmetic by Richard Kaye, Roman Kossak, and Tin Lok Wong
    9. On the Notion of Proving its own Consistency by Doukas Kapantais
    10. Randomness, pseudorandomness and models of arithmetic by Pavel Pudlak
    11. The asymptotic behaviour of the number of trees in certain classes by Jan-Christoph Schlage-Puchta
    12. What is Sequentiality? by Albert Visser
  • II. Some problems in Logic and Number Theory, and their connections Thesis (1981) of Alan Robert Woods
    Note by Costas Dimitracopoulos
  • III. Models of Arithmetic Thesis (1978) of H. Lessan
    Note by George Wilmers

September 2013

ISBN (Paperback): 9781575867236
ISBN (electronic): 9781575867243

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