  Tarski considers the possibility of adding an ω-rule to the accepted rules of inference, that is, a rule which allows us to deduce a universal sentence of the form of A from the set of sentences A0, A1, etc. However, he says that such a rule would differ in very essential respects from the old rules: it is not a finitary rule, while all the accepted rules in common deductive systems are finitary. Also, Tarski immediately takes importance away from the further suggestion of supplementing the old system of rules by means of a formalized finitary ω-rule. He points out that in view of Gödel’s incompleteness results, no matter how many new finitary rules or axioms we add to certain higher-order theories, they will still remain incomplete theories, in fact ω-incomplete theories. This discussion is enough to show that “in order to obtain the proper concept of consequence, which is close in essentials to the common concept, we must resort to quite different methods and apply quite different conceptual apparatus in defining it” (Tarski 1983c, p. 413). The different methods and the different conceptual apparatus that Tarski has in mind are going to be “the methods which have been developed in recent years for the establishment of scientific semantics, and the concepts defined with their aid” (Tarski 1983c, p. 414; a footnote refers us to Tarski’s monograph on truth).