Abstract

One of the famous achievement of contemporary logic was K. Gödel`s argumentation that the axiomatic systems which contain arithmetic, , if they are consistent, then they are also indecidable. This result was interpreted against logicism, as the foundation of mathematics program, which proposes just to embed the arithmetic in a logical system.The decidability is the property of an axiomatic system in which, for every well formed expression in that system , we may establish if it is a theorem of the system or not, using only the axioms and the derivability rules accepted in S. Gödel`s indecidability theorem shows that in such a system, we may always find a formula about which it`s impossible to decide if it is theorem or not. Of course, the problem appears only in the case of consistent axiomatic systems because, for an inconsistent system, any formula is a theorem