Studies in Weak Arithmetics

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