T-schema deflationism versus Gödel's first incompleteness theorem (Q2765279)

The author uses an extension of an argument by \textit{V. McGee} [J. Philos. Log. 21, 235-241 (1992; Zbl 0773.03003)] in order to show that a theory of truth based on the T-schema restricted to suitable instances runs into problems with recursive axiomatizability. Thus, the author argues, T-schema deflationism, which advocates such a theory, fails. NEWLINENEWLINENEWLINERoughly the argument runs as follows: According to McGee's theorem every arithmetical sentence \(\phi\) is equivalent to an instance of scheme T because \(\phi\leftrightarrow (T[\gamma] \leftrightarrow \gamma)\) is logically implied by the instance \(\gamma\leftrightarrow (T[\gamma] \leftrightarrow \phi)\) of the diagonal lemma. If the theory of truth comprises those instances of schema T that are equivalent to true arithmetical sentences, but does not include any instances equivalent to any false arithmetical sentences, then the set of all arithmetical truths is recursive in the truth theory, which therefore cannot be recursively enumerable. Hence one cannot have a recursively enumerable theory of truth containing an instance of schema T that is equivalent to an arithmetical sentence \(\phi\) if and only if \(\phi\) is true.