On the sense of beauty in Mathematics and what showed to be beautiful for us in the proof of Gödel's incompleteness theorems

Abstract

In this paper we present a study that had as a guiding question: What is the beauty in mathematics and the beauty in Gödel's incompleteness theorem? We seek to present identifiable content for what we understand as beauty in Gödel's incompleteness theorem. For this, a bibliographic study was made and different notions of beauty in mathematics are brought and articulated. We also present our understanding of beauty in Gödel's theorem. We understand that the meaning of mathematical beauty of a theorem is that of an illumination that evidences the result. Moreover, we understand that this light allows itself to be seen insofar as one is familiar with the theory and with the tools used in the demonstration, to the point of being able to perceive the axioms used, the conciseness of the proof, the originality of the articulation of ideas, the possibilities of generalizing the result and the opening of new research fronts. We also recognize that the knowledge built by Gödel in the elaboration of his completeness theorem was fundamental in the view of the problem of the consistency of arithmetic and in the approach chosen to demonstrate the consistency of arithmetic, which became the demonstration of the incompleteness of the theory of arithmetic.