  Russell communicated this problem to Frege, and his letter, together with Frege’s answer appear in van Heijenoort 1967. It is important to realise that the formulation (*) does not apply as it is to Frege’s system. As Frege himself wrote in his reply to Russell, the expression “a predicate is predicated of itself” is not exact. Frege had a distinction between predicates (concepts) and objects. A (first-order) predicate applies to an object but it cannot have a predicate as argument. The exact formulation of the paradox in Frege’s system uses the notion of the extension of a predicate \(P\), which we designate as \(\varepsilon P\). The extension of a predicate is itself an object. The important axiom V is: