On Tarski's Undefinability Theorem

Abstract

This paper shows that Tarski's revised Undefinability Theorem obscures a liar paradox affecting various systems in scope. Consider for example Tarski's general theory of classes (which contains only variables of finite order) as the object system (O), and a metatheory M which has a transfinite variable (Tr^{omega}) such that (under the intended interpretation): x in Tr^{omega} holds iff the O formula named by x is true. As there are only a denumerable number of true O formulae, it is provable within M, if we assume the axiom of choice, that there exists a class (X^{4}), named, under the intended interpretation, by a variable of finite order, that contains all, and only, the classes corresponding to the Godel numbers of the true O formulae. If the class assigned to Tr^{omega} is well defined we may conservatively extend M to M' such that: M' contains a new constant of finite type (Tr^{4}) and proper axioms assigning the class X^{4} to Tr^{4}. It is easily shown that M' exhibits a liar paradox. The proof holds, with appropriate changes, for ZFC and also implies that standard proofs that "arithmetic truth is not arithmetic" beg the question of whether the standard interpretation of first-order arithmetic is well defined.