 It’s worth mentioning, firstly, that this principle underlies diagonal argumentation in general (cf. Gaifman 2006). Even venerable examples such as Post’s informal argument that there is a recursively enumerable set of positive integers whose complement is not recursively enumerable, rely in essence on  CL  (Davis 1965: 312). A slight variant of CL is frequently found in the literature on undecidability (cf. Shoenfield 1967: 131). Secondly, the proof of CL does not rely essentially on any axiom of set theory, and it is therefore a theorem of (first order) logic (just as the theorem asserting the non-existence of the set of non-self-membered sets does not rely on any axiom of set theory). Of course, the famous corollary of CL about cardinality relies essentially on axioms of set theory—in particular the power set axiom. But the theorem that there can be no one-one function from a set onto its power set is otherwise a direct consequence of CL. In this case, \(D_f\) must be in \(B\), since it belongs to the power set of \(A\), and the conclusion must be that \(f\) does not exist. Third, Grelling’s paradox as well as Berry’s paradox can be used as a basis for alternative proofs of the first incompleteness theorem (Boolos et al. 2007; Kripke 2014). Lastly, a somewhat more general form of CL pertaining to relations in general—not just to functions—, which is now accepted as a version of Cantor’s Theorem, was already noticed by Russell (1903). Whereas CL is essentially a principle of first order logic, the relational version is second order in nature.