The Notion of Cardinality and the Continuum Hypothesis

The present chapter will discuss Cantor’s first two articles on topics that would become the core of transfinite set theory-his famous work on the non-denumerability of the reals [1874] and the equipollence of continua of any number of dimensions [1878]. This was the birth of the notion of cardinality or power of an infinite set, 2 which Cantor presented to the public in the second paper, an epochmaking article which also contained his first version of the Continuum Hypothesis. The paper that might be regarded as the first published contribution to transfinite set theory [Cantor 1874] appeared under complex circumstances, and it was far from offering a clear idea of Cantor’s actual views. The crucial idea that infinite sets have different powers had been born for him, but not for most of his readers. Apparently due to the influence of Weierstrass, his presentation failed to emphasize that point. Cantor’s abstract viewpoint and the notion of power only became clear with his second paper [1878], which makes several references to the first one in order to place it in a new light [ op.cit. , 120, 126]. Cantor even felt the need to reformulate his 1874 proof five years later [Cantor 1879/84, part I], which underscores the peculiar nature of the 1874 article. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.