 Although it is conceivable that Cantor might have found that paradox as early as 1883, immediately after introducing the transfinite ordinals (for arguments in favour of this idea see Purkert &amp; Ilgauds 1987 and Tait 2000), the evidence indicates clearly that it was not until 1896/97 that he found this paradoxical argument and realized its implications. By this time, he was also able to employ Cantor’s Theorem to yield the Cantor paradox, or paradox of the alephs: if there existed a “set of all” cardinal numbers (alephs), Cantor’s Theorem applied to it would give a new aleph \(\aleph\), such that \(\aleph &lt; \aleph\). The great set theorist realized perfectly well that these paradoxes were a fatal blow to the “logical” approaches to sets favoured by Frege and Dedekind. Cantor emphasized that his views were “in diametrical opposition” to Dedekind’s, and in particular to his “naïve assumption that all well-defined collections, or systems, are also ‘consistent systems’ ” (see the letter to Hilbert, Nov. 15, 1899, in Purkert &amp; Ilgauds 1987: 154). (Contrary to what has often been claimed, Cantor’s ambiguous definition of set in his paper of 1895 was intended to be “diametrically opposite” to the logicists’ understanding of sets—often called “naïve” set theory, which could more properly be called the dichotomy conception of sets, following a suggestion of Gödel.)