 Cantor’s paradoxes convinced Hilbert and Dedekind that there were important doubts concerning the foundations of set theory. Hilbert formulated a paradox of his own (Peckhaus &amp; Kahle 2002), and discussed the problem with mathematicians in his Göttingen circle. Ernst Zermelo was thus led to discover the paradox of the “set” of all sets that are not members of themselves (Rang &amp; Thomas 1981). This was independently discovered by Bertrand Russell, who was led to it by a careful study of Cantor’s Theorem, which conflicted deeply with Russell’s belief in a universal set. Some time later, in June 1902, he communicated the “contradiction” to Gottlob Frege, who was completing his own logical foundation of arithmetic, in a well-known letter [van Heijenoort 1967, 124]. Frege’s reaction made very clear the profound impact of this contradiction upon the logicist program. “Can I always speak of a class, of the extension of a concept? And if not, how can I know the exceptions?” Faced with this, “I cannot see how arithmetic could be given a scientific foundation, how numbers could be conceived as logical objects” (Frege 1903: 253).