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