La critique de la théorie des ensembles dans la dissertation de Brouwer (1907)

Mathématiques et sciences humaines. Mathematics and social sciences

Bourdeau, Michel¹ 

关键词: intuitionism;    transfinite;    continuum;    ordinals;   

DOI  :  10.4000/msh.2892

学科分类：数学（综合）

来源: College de France * Ecole des Hautes Etudes en Sciences Sociales (E H E S S)

【 摘 要 】

Although Brouwer had not developed intuitionistic mathematics by 1917, some of his main ideas are already to be found in his 1907 Dissertation, for instance his attitude towards set theory. We are invited to distinguish two aspects in Cantor's creation: the authentic mathematical contribution (topology, ordinals), and a spurious one, which leads to contradictions and reflects an excessive confidence in the power of logic. Consequently, the transfinite has two faces: the hierarchy of the alephs, and the ordinals. Brouwer agrees with the latter, not with the former. This explains also why the Continuum Hypothesis is discussed in two different places: the mathematical version in the first chapter, the logical one in the third chapter. In the latter, Brouwer accepts the two generating principles for ordinals, but he thinks that these provide no ground for taking the second class of numbers as a completed totality.

【 授权许可】

Unknown