The diagonal lemma as the formalized Grelling paradox 1

György Serény Gödel's diagonal lemma (which is often referred to as fix-point or self-referential lemma) summarizes very succinctly the ability of first-order arithmetic to 'talk about itself', a crucial property of this system that plays a key role in the proof of Gödel's incompleteness theorem and in those of the theorems of Tarski and Church on the undefinability of truth and undecidability of provability respectively. In fact, with the representability of provability at hand, these three main limitative theorems of logic can be considered to be simple applications of the lemma (cf. e.g. [2] pp. 227–231). Due to this central role, the proof of the lemma could shed light on the very essence of these fundamental theorems. In spite of the fact that it is common knowledge that Gödel's proof of the incompleteness theorem is closely related to the Liar paradox, the proof of the lemma as it is presented in textbooks on logic is not self–evident to say the least. Indeed, in the Handbook of Proof Theory, the proof of the lemma is introduced by the following remark (see [1], p.119): 'This proof [is] quite simple but rather tricky and difficult to conceptualize.' Or to quote another opinion, 'The brevity of the proof does not make for transparency; it has the aura of a magician's trick' (cf. [4], p. 1). It seems, therefore, that the words of a respected logician reflect a widespread attitude to the proof of the lemma (see [5]): '[This] result is a cornerstone of modern logic. [...] You would hope that such a deep theorem would have an insightful proof. No such luck. [...] I don't know anyone who thinks he has a fully satisfying understanding of why the Self-referential Lemma works. It has a rabbit-out-of-a-hat quality for everyone.' In view of these remarks, we think that it is worth drawing attention to a possibility of making the proof of the lemma completely transparent by showing that it is simply a straightforward translation of the Grelling paradox into first-order arithmetic. 2 Notation Our formal language is that of first-order arithmetic. Q stands for Robinson arithmetic while ω is the set of natural numbers. g is any one of the standard Gödel numberings and F m n is the set of formulas with all free variables among the first n ones. For the sake of simplicity, we shall denote the closed terms corresponding to natural numbers …