Diagonal-free proofs of the Diagonal Lemma

The Diagonal Lemma (of Gödel and Carnap) is one of the fundamental results in Mathematical Logic. However, its proof (as presented in textbooks) is very un_intuitive, and a kind of “pulling a rabbit out of a hat”. As a matter of fact, several theorems that are proved by using this lemma now have diagonal-free proofs. One example is Gödel's Incompleteness theorem for which several diagonal-free proofs are given by Kleene, Chaiting and Boolos; another example is Tarski's Undefinability theorem for which Robinson and Kotlarski gave diagonal-free proofs.
In this talk we will see that a weak form of the Diagonal Lemma is equivalent with Tarski's theorem, and we will explore some different proofs for these theorems. Even though there will be no new theorem in this talk, several interesting proofs for old theorems will be presented.