First-order Peano arithmetic and Tarski’s elementary geometry are quite different formal theories. The first one is incomplete and (essentially) undecidable whereas the second is complete and decidable. In this talk we point to other less well known properties and distinctions concerning formal theories of arithmetic and geometry. We will specially look at Raphael Robinson’s theory of arithmetic Q, a very weak theory without induction. Work due to Nelson, Solovay, Hajek, Buss, Wilkie, Paris, Pudlak, Visser, Friedman, Ferreira, etc. shows that there are very interesting theories which Robinson's Q is able to interpret. For instance, Q is able to interpret Tarski’s elementary geometry. In order to discuss this, we make a detour through weak analysis and the work of A. Fernandes, F. Ferreira and G. Ferreira.
