Undefinability vs. definability of satisfaction and truth

Roman Murawski

pp. 203-215

in: Jan Woleński, Eckehart Köhler (eds), Alfred Tarski and the Vienna circle, Berlin, Springer, 1999

Abstract

Among the main theorems obtained in mathematical logic in this century are the so called limitation theorems, i.e., the Löwenheim-Skolem theorem on the cardinality of models of first-order theories, Gödel's incompleteness theorems and Tarski's theorem on the undefinability of truth. Problems connected with the latter are the subject of this paper. In Section 1 we shall consider Tarski's theorem. In particular the original formulation of it as well as some specifications will be provided. Next various meanings of the notion of a satisfaction predicate will be studied. In Section 2 the problem of definability of the notion of truth, in particular of the notion of truth for the language of Peano arithmetic PA, will be discussed. It will be explicitly shown that the notion of satisfaction (and consequently also the notion of truth) for the language of PA can be defined in a certain weak fragment of the second order arithmetic. Finally the axiomatic characterization of satisfaction and truth (i.e., the notion of a satisfaction class) as well and its mathematical and philosophical meaning will be discussed.