The fact that the rule of mathematical induction is contradictory with the rest of clauses used by G?del to prove his undecidability and incompleteness theorems is proved in this paper. This means that those theorems are invalid.
In section 1, a study is carried out on the mathematical induction principle, even though it is not directly relevant to the problem, just to familiarize the reader with the operations that are used later; in section 2 the rule of mathematical induction is introduced, this rule has a metamathematical character; in section 3 the original proof of G?del's undecidability theorem is reproduced, and finally in section 4 the same proof is given, but now with the explicit and formal use of all the axioms; this is needed to be able to use logical resolution. It is shown that the inclusion of the mathematical induction rule causes a contradiction.
