 The reason for the inconsistencies was eventually clarified by Curry’s 1941 essay. There he distinguishes two basic completeness notions: a system \(S\) is deductively complete, if, whenever it derives a proposition \(B\) from the hypothesis \(A\), then it also derives the implication \(A \rightarrow B\) (deduction theorem or introduction rule for implication); \(S\) is combinatorially complete (Curry 1941, p. 455.) if, whenever \(M\) is a term of the system possibly containing an indeterminate \(x\), there exists a term (Church’s \(\lambda x.M)\) naming the function of \(x\) defined by \(M\). Curry then remarks that the paradox of Kleene-Rosser arises because Church’s and Curry’s systems satisfy both kinds of completeness, thus showing that the two properties are incompatible. In the more technical part of the paper, Curry carefully axiomatizes the main ingredients exploited by Kleene and Rosser and carries out a lot of non-trivial work both on the logical side and the mathematical side (e.g, to develop a portion of recursive arithmetic, to define the existence of an enumerator, i.e., a term \(T\) such that, if \(a\) is the Gödel number of a closed term \(M\) and \(Z_a\) is the term formally representing \(a\), then \(TZ_a =M\) is provable in \(S\), etc.). Curry’s proof of the inconsistencies of combinatory systems is unsatisfactory because it heavily uses a detour through number theory and Gödelization, which, as a matter of fact, is unnecessary as Curry himself soon discovered and presented in a paper with the same title as the one by Kleene and Rosser, “in deference to the original discoverers of the contradiction” (Curry 1942). The main result in it is the following theorem (Curry’s paradox, see the entry  Curry’s Paradox):