 As the reactions of the mathematical world made clear, the paradoxes were crucially involved around 1905 when basic problems of set theory were worked on. Indeed, the new contradictions not only affected the conception of set and logical concepts, they also came to include the notion of definability and its relation with a fundamental issue: the structure of the mathematical continuum and in particular whether the continuum can be well-ordered and whether Cantor’s Continuum Hypothesis (CH) holds.