The development of logical-mathematical instruments and concepts of Kurt Gödel's First Incompleteness Theorem - Bório de Medeiros, Bismarck (2024)
The article seeks to elucidate the investigations and advances in Mathematics and Logicassociated with philosophical conceptions that culminated in Kurt Gödel's First Incompleteness Theorem. For this, we will make a historical and conceptual approach to Mathematics from the second half of the 19th century to the first half of the 20th century, indicating elements and mathematical instruments developed to solve problems, as well as philosophical assumptions and commitments that accompany activities aimed at the formalization and foundation of contemporary mathematical logic that helped Gödel to elaborate his demonstration and clarify the limitations of formal systems with a minimum of Arithmetic. In this way, we will deal with how the problems from the establishment of non-Euclidean geometries and the Set Theory culminated in different lines of research focused on the foundations of Mathematics, as well as the discovery of paradoxes and the controversial notion of the Infinite demanded finitary and recursive methods, such as instruments created for mathematical demonstrations in this period helped in the emergence of metamathematics until Gödel's proof. At the end, we will make a general synthesis and reflection on this intellectual enterprise in the progress of mathematical investigation itself.