The years up to 1914 were a crucial period of diffusion and recognition for set theory. During the 1890s the new vision of mathematics and the Cantorian ideas spread out, while the 1900s saw fundamental new contributions in the hands of a new generation of cultivators - Zermelo and Hausdorff above all. But this was also a period of heated debates surrounding the notion of arbitrary set and its expressions, the Axiom of Choice and the Well-Ordering Theorem. It was also the time in which the paradoxes emerged, heralded by Russell. Thus, the diffusion of set theory was accompanied by much ambivalence and confusion. The acceptability of abstract mathematics was in question, as were the relations between logic and set theory.