 Russell was at first entangled in the study of “the contradictions in the relation of continuous quantity to number and the continuum” (cited in Moore 1995, p. 219), and he obtained his contradiction (May 1901) as a result of considering the antinomy arising with Cantor’s theorem (see Russell 1903, footnote 7, par. 344; par. 100, p. 101). Russell probably realized the importance of the discovery only after Frege’s reply. The effect of the antinomy is that it is impossible to have an abstraction operation \(\phi \mapsto \{x \mid \phi \}\) mapping injectively any concept (property) \(\phi\) into its extension (the class of all \(x\) such that \(\phi(x))\) (i.e., so that if the classes defined by \(\phi\) and \(\psi\) are equal, then \(\phi(a)\leftrightarrow \psi(a)\), for every object \(a)\). As a consequence, it is also impossible to lay out the foundations of set theory on a pure logical notion of set where membership faithfully mirrors predicate application, in the sense that, in the light of Frege, \(x \in y\) means that (1) \(y = \{x \mid \phi(x)\}\) for some concept \(\phi\), and (2) \(\phi\) truly applies to \(x\). (For historical details on Russell’s discovery of the paradox, see Moore 1995).