Godel, Tarski, Church, and the Liar

Abstract

The fact that the famous Godel incompleteness theorem and the archetype of all logical paradoxes, that of the Liar, are related closely is, of course, not only well known, but is a part of the common knowledge of logician community. Actually, almost every more or less formal treatment of the theorem (including, for that matter, Godel's original paper as well) makes a reference to this connection. In the light of the fact that the existence of this connection is a commonplace, all the more surprising that very little can be learnt about its exact nature. Now, it emerges from what we do in this paper that the general ideas underlying the three central limitation theorems of mathematics, those concerning the incompleteness and undecidability of arithmetic and the undefinability of truth within it can be taken as different ways to resolve the Liar paradox. In fact, an abstract formal variant of the Liar paradox constitutes a general conceptual schema that, revealing their common logical roots, connects the theorems referred to above and, at the same time, demonstrates that, in a sense, these are the only possible relevant limitation theorems formulated in terms of truth and provability alone that can be considered as different manifestations of the Liar paradox. On the other hand, as illustrated by a simple example, this abstract version of the paradox opens up the possibility to formulate related results concerning notions other than just those of the truth and provability.