 where \(\psi\) is a sentence of \(L\) and \(s\) is the canonical name for the Gödel number of \(\psi\). Padoa’s method implies that \(\mathbf{H}\) is not admissible for defining \(Tr\). For \(\mathbf{H}\) does not fix the interpretation of \(Tr\) in all interpretations of \(L\). In particular, it does not do so in the standard model, for \(\mathbf{H}\) places no constraints on the behavior of \(Tr\) on those numbers that are not Gödel numbers of sentences. (If the coding renders each natural number a Gödel number of a sentence, then a non-standard model of Peano Arithmetic provides the requisite counterexample: it has infinitely many expansions that are models of \(\mathbf{H}\).) A variant of this argument shows that Tarski’s theory of truth, as formulated in \(L^{+}\), is not admissible for defining \(Tr\).