Aspectos do primeiro teorema da incompletude de Kurt Gödel e uma análise da prova teorético informacional de Gregory Chaitin

Abstract

The work seeks to elucidate and understand relevant aspects in the structure of paradoxical undecidable sentences in consistent formal systems that contain Dedekind-Peano Arithmetic. The first chapter exposes the investigations and advances in Mathematics and Logic associated and the philosophical conceptions that culminated in Kurt Gödel's First Incompleteness Theorem, published in his article Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, in 1931. 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 with its main lines of thought, indicating the mathematical elements and instruments developed to solve certain problems, as well as philosophical assumptions and commitments that accompanied the activities aimed at the formalization and foundation of contemporary Mathematical Logic that helped Gödel to elaborate his demonstration and to explain limitations of such formal systems. The second chapter aims to analyze the components and expose or elaborate formalized undecidable sentences based on paradoxes considered epistemic or semantic. Will be discussed paradoxes expressed implicitly and explicitly in the structure of undecidable sentences. We approaching similarities and distinctions of both finite and infinite undecidable sentences, seeking to understand the proofs and phenomena that lead to the incompleteness of formal systems that contains Dedekind- Peano Arithmetic. Soon after, the third chapter will focus on the application of Algorithmic Information Theory developed by Gregory Chaitin to demonstrate a discussed version of incompleteness of formal systems based on Berry's Paradox. The critical literature on this information-theoretic version will be resumed, as well as an analysis based on the sentences seen above, carrying out a scrutiny of the justifications and definitions used in Chaitin's proof. At the end, we open a discussion about the nature of incompleteness associated with the notion of computability and the limits of finite mechanical processes.