Resumen

DA SILVA, Ricardo. The Gödel incompleteness theorems, set theory and the programme of David Hilbert. EPISTEME [online]. 2014, vol.34, n.1, pp.19-40. ISSN 0798-4324.

Gödel proved in 1931 that for any formal recursive system Z powerful enough to derive the Peano axioms and also supposed to be consis­tent, we have that in the System there are undecidable propositions, i.e., the system is not complete. Moreover, Gödel proved that if the Z system is con­sistent then it can not derive in Z a proposition asserting the consistency of Z. These results are known as Gödel's First Incompleteness Theorem and Gödel's Second Incompleteness Theorem. Such results have a great impact on the in­vestigation of the foundations of mathematics that had been developing in the first thirty years of the last century, and it, furthermore, has implications for philosophy of mathematics of that time. This article is structured in three parts: In the first part we deal with the formulation of the incompleteness theorems and the main ideas of its proof in each case. Then, we will show an application of the Second Incompleteness Theorem in set theory concerning inaccessible cardinal. Finally, we will develop the philosophical consequences that Gödel's incompleteness theorems have on the meta-mathematical project that David Hilbert proposed.

Palabras clave : Incompleteness; inaccessible cardinal; Hilbert.

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