i have tried to prove, in first order logic, that every satisfiable sentence (without function symbols) with at most two variables has a finite model. my attempts were unsuccessful. this is an exercise from the wider model thoery book written by Hodges (ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS, Volume 42 - model theory, page 111). It follows an exercise about Immerman's pebble game, probably as an application. it's easy to see that proving the following problem will suffice: given a structure A, prove that for every number n, there is a finite structure B such that player II has a winning strategy in immerman's pebble game of length n with 2 pebbles. thanks for the help =]