A rejeição do princípio do terceiro excluído e suas consequências na aritmética de Heyting

Detalhes bibliográficos
Ano de defesa: 2014
Autor(a) principal: Jackeline Nogueira Paes
Orientador(a): Não Informado pela instituição
Banca de defesa: Não Informado pela instituição
Tipo de documento: Dissertação
Tipo de acesso: Acesso aberto
Idioma: por
Instituição de defesa: Universidade Federal de Minas Gerais
UFMG
Programa de Pós-Graduação: Não Informado pela instituição
Departamento: Não Informado pela instituição
País: Não Informado pela instituição
Palavras-chave em Português:
Link de acesso: http://hdl.handle.net/1843/BUOS-9QHJ9B
Resumo: During the transition from the XIX to the XX century, mathematics got important developments. However, these developments were followed by the discovery of paradoxes, including the known paradoxes of Russell and Cantor, who opened the discussion on the foundations of mathematics. These discussions aimed to find a secure foundation free of errors and impreciseness for this science. The Intuitionism, an alternative to classical mathematics, was originated in the constructivist ideas exposed by the Dutch mathematician L. E. J. Brouwer, and has resulted in the rejection of the Principle Excluded Middle. This rejection is mainly based on the thesis that mathematical objects are constructs of the mind and, therefore, the rejection of one supersensible and preexisting field of mathematics entities. We aim in this work to demonstrate which that thesis depends on the way in which Brouwer includes three fundamental concepts to the mathematics, namely, infinite, truth and existence. After, we demonstrate the consequences of the rejection of Excluded Middle in arithmetic. We will see, at least in the case of arithmetic that rejection of the Excluded Middle would not be enough to avoid inconsistencies. We also discuss the translations of double negations performed by Kolmogorov, Gentzen and Gödel and, finally, we will introduce a proof of the equiconsistency of intuitionistic and classical arithmetical.