 For the proof, it is enough to find, for any given term B, a term A such that \(A=A\supset B\). Curry notes that a twofold construction is possible. By direct self-reference, we can choose: \(A = HH\), where \(H=\lambda Y (N(YY))\) and \(N = \lambda X (X\supset B)\). On the other hand, one can apply indirect self-reference and exploit the machinery of Curry 1941 and Kleene-Rosser: using an enumerator \(T\), one sets \(U = \lambda X (TXX \supset B)\) and \(A=UZ_u\), being \(u\) the Gödel number of \(U\). Curry suggests that these two routes are akin, respectively, to Russell’s paradox and to the Liar. It is interesting to note that the two ways correspond to by-now standard tools, the so-called first fixed point theorem and second fixed point theorem of combinatory logic and lambda calculus (Barendregt 1984, p. 131 and p. 143; see also the distinction between the first recursion theorem and the second recursion theorem of classical recursion theory, in this SEP, entry  recursive functions).