THE IMPORTANCE OF TEACHING GÖDEL'S INCOMPLETENESS THEOREM IN MATHEMATICS TEACHER EDUCATION.

In this article we discuss our views on the importance of teaching Gödel's incompleteness theorem (GIT) in Mathematics teaching degrees. This theorem addresses formal Mathematics systems, such as Peano arithmetic. It does not directly deal with elementary school contents taught by Mathematics teachers. It highlights a fundamental characteristic of the axiomatic method through which Mathematics is produced. The main ideas established by GIT cover the incompleteness of proof of arithmetic consistency in arithmetical language. Incompleteness means the existence of true propositions about natural numbers that cannot be proved by elementary arithmetic. A corollary established that, if arithmetic is consistent, such consistency cannot be proved by any metamathematical argument that can be represented according to the formalism of arithmetic. In other words, not even the most elementary axiomatic systems of Mathematics, which formalize natural numbers, can prove all truths established in its own spectrum. Knowing this result and the conclusions that it reaches is essential in terms of the mathematical culture of a teacher preparing to teach Mathematics in schools, as a means to avoid nurturing the notion of complete Mathematics and the totalitarianism of the axiomatic method. [ABSTRACT FROM AUTHOR]