Gödel on Tarski

Abstract

Contacts of the two logicians are listed, and all Gödel's written mentions of Tarski's work are quoted. Why did Gödel almost never mention Tarski's definition of truth in his notes and papers? This puzzle of Gödel's silence, proposed by Feferman, is not merely biographical or psychological but has interesting connections to Gödel's philosophical views. No satisfactory answer is given by the three “standard” explanations: (i) no need to repeat the work already done; (ii) Tarski's achievement was obvious to Gödel; (iii) Gödel's exceptional caution. In fact, (i), Tarski had done the work, but Gödel almost never mentioned the achievement; (ii), the obviousness is no explanation for the omission of Tarski's work in contexts in which an application of the definition of satisfaction was useful, and even necessary; (iii), the point was not just caution: if Gödel had felt the need to mention the program of scientific semantics he could easily have done that in his manuscripts, or in conversations. Three ideas, detectable in Gödel's approach, can help us understand Gödel's silence: (A) the idea of truth as the intuitive provability in the most general sense; defining it set-theoretically would contribute nothing. (B) the idea of truth as an inexhaustible idea in the sense of Kant; “truth in general” is a category that must be applicable to all kinds of sentential expressions; also, while for Gödel language was secondary, Tarski's definition is focused on language. (C) the idea of logic as the universal language, in Hintikka's sense, as opposed to the perception of logic as a reinterpretable calculus; hence the thesis that semantics is inexpressible. Gödel always remains a Platonist who asks a natural question: what does really happen in the realm of abstract objects?