On Tarski's Undefinability Theorem
Abstract
This paper shows that Tarski's revised Undefinability Theorem fails to avoid a liar-like paradox for systems of infinite order. Suppose for the general theory of classes, for transfinite Tr of the metatheory M, x in Tr holds iff the object system (O) formula named by x is true. By instances of the comprehension axiom for the system M, it is provable within the system, if we assume the axiom of choice, that there exists a class that contains all, and only, the classes corresponding to the Godel numbers of the true O formulae (whose names are members of Tr). If the class assigned to Tr under the intended interpretation is well defined, then we may add proper axioms, corresponding to these comprehension axioms, defining a new constant of finite type, Tr4, so that n in Tr4 holds - for n an M-expression for the nth natural number - iff the O-formula with this Godel number is true. Plainly however we may deduce from these axioms Tr-free theorems which correspond to Convention T biconditionals. From these the original liar-like paradox is again obtained. As a corollary, it may also be seen that standard proofs that "arithmetic truth is not arithmetic", beg the question of whether the standard interpretation of first-order arithmetic is well defined.