Let \(\mathbb N=(\omega,+,\times,<,0,1)\) be the standard model of Peano Arithmetic (PA). Tarski's theorem on the undefinability of truth claims that the set TA \(=\{\phi: \mathbb N \models \phi\}\) is undefinable in \(\mathbb N\). \textit{A. Robinson} gave a model-theoretic proof of Tarski's theorem [Nagoya Math. J. 22, 83--117 (1963; Zbl 0166.26101)]. In the paper the author discusses Robinson's proof of Tarski's theorem and gives two other ``diagonal-free'' proofs. In Notre Dame J. Formal Logic 36, 519--530 (1995; Zbl 0848.03016), the author posed a problem about a possible converse to Tarski's theorem. In this paper he presents some arguments in favor of a negative answer to this question. Finally, the author poses another open problem: Is there an easy proof of Ehrenfeucht's lemma from Tarski's theorem?