Incompleteness, Undecidability and Platonism in the Work of Kurt Gödel

• Autores: Joan Roselló
• Localización: VII Conference of the Spanish Society for Logic, Methodology and Philosophy of Science: Santiago de Compostela, Spain, 18-20 July 2012 / Sociedad de Lógica, Metodología y Filosofía de la Ciencia en España (aut.), Concepción Martínez Vidal (dir. congr.), José L. Falguera López (dir. congr.), José Miguel Sagüillo Fernández-Vega (dir. congr.), Víctor Martín Verdejo Aparicio (dir. congr.), Martín Pereira Fariña (dir. congr.), 2012, ISBN 978-84-9887-939-1, págs. 74-80
• Idioma: inglés
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• Resumen

  • This article analyses the relationship between some mathematical and philosophical implications of Gödel’s incompleteness theorems, particularly the phenomenon of the inexhaustibility of mathematics and the possible existence of absolutely undecidable sentences, and his Platonism or conceptual realism. In this regard, we conclude that Gödel’s proposal to extend Zermelo-Fraenkel’s axiomatic set theory with the so-called axioms of infinity, through which he sought to prove hitherto undecidable sentences in set theory and number theory, is closely related not only to his understanding of the phenomenon of incompleteness, but also with his Platonic view on set theory and mathematics in general. The article also aims to describe the development of Gödel’s philosophical ideas about the nature of mathematics and their relationship with other foundational programs or philosophical approaches to mathematics such as Carnap’s syntactical program, Hilbert’s finitism or Husserl’s phenomenology. Regarding this, we conclude that Gödel’s late interest in Husserl’s transcendental idealism responds to the fact that it offers a general philosophical framework for some of Gödel ideas in the philosophy of mathematics and a method for the search and justification of new axioms in mathematics.