Observations concerning Gödel’s 1931

This article demonstrates the invalidity of Theorem VI in Gödel’s 1931, showing that propositions (15) and (16), derived from the definition 8.1, in its proof, are false in PA. Introduction. Developed as a consequence of the crisis of the foundation of mathematics due to the discovery of the antinomies, Hilbert’s formalism planned as criterions of adequacy for the axiomatic systems, the achievement of their coherence and completeness [8][9]. The result of incompleteness for any system embodying the arithmetic of the positive integers, obtained by Gödel in 1931, grafted on Hilbert’s program as the establishment of its definitive unattainableness [5][7]. This article exposes proof of the invalidity of the so-called Gödel’s first incompleteness theorem, according which it is possible to construct a statement that, although true, turns out to be nor provable nor refutable for the system. This is obtained in two steps: defining refutability within the same recursive status as provability and showing that propositions (15) and (16), derived from the definition 8.1 in Gödel’s 1931 article ([5] 174-175), are false and therefore unacceptable for the system. The achievement of their falsity blocks the derivation of theorem VI, which turns out to be therefore invalid, together with all the depending theorems. For many mathematicians of the end of the nineteenth-century, the condition for a theory to be no-contradictory was considered to be sufficient to the existence of the objects of the theory. The result of Gödel in 1930, assuring the existence of the model for the first order logic, appeared to be a confirmation of such position. But, for Gödel, the identification of the no-contradictoriness with the existence seemed “manifestly presupposes the axiom that every mathematical problem is solvable. Or, more precisely, it presupposes that we cannot prove the unsolvability of any problem.” ([3] 60-61) While, thorough Gödel’s contributions, it is recurrent the existence of unsolvable problems in mathematics (see note 61 in [5] 190-191) and its underlying interchange with the existence of undecidable proposition in the formal system ([5] 144145). This interconnection of unsolvable problems in mathematics with undecidable propositions in the formal system, was supported by the idea that “it is conceivable that there are true propositions (which may even be provable by means of other principles) that cannot be derived in the system under consideration.” ([4] 102-103) So that for Gödel there were unsolvable problems in mathematics that, although contentually true, were unprovable in the formal systems. “(Assuming the consistency of classical mathematics) one can even give examples of propositions (and in fact of those of the type of