Non-finite axiomatizability of first-order Peano Arithmetic

Abstract

[en] The system of Peano Arithmetic is a system more than enough for proving almost all statements of the natural numbers. We will work with a version of this system adapted to first-order logic, denoted as PA. The aim of this work will be showing that there is no equivalent finitely axiomatizable system. In order to do this, we will introduce some concepts about the complexity of formulas and codification of sequences to prove Ryll-Nardzewski’s theorem, which states that there is no consistent extension of PA finitely axiomatized.