Semantics and Truth by Jan Woleński

Author:Jan Woleński
Language: eng
Format: epub
ISBN: 9783030245368
Publisher: Springer International Publishing

As far as metamathematics is concerned, Tarski summarized the situation in the following statement (Tarski 1954, p. 713) :As an essential contribution of the Polish school to the development of metamathematics one can regard the fact that […] it admitted into metamathematical research all fruitful methods, whether finitary or not.

And here we have the point where semantics and metamathematics meet, so to speak, on the official level. It is, of course, not true that truth did not belong to mathematical jargon before Tarski . It was used informally, although essentially—when the completeness problem was being formulated, for example. However, as Gödel remarked, it was not formally elaborated, although he employed it himself in stating the completeness theorem (see Gödel 1930) , and in informal explanations the first incompleteness theorem (see Chap. 8, Sect. 8.​4). It was Tarski, among others, who quite consciously combined philosophical and mathematical interest in the precise elaboration of semantic concepts. It was accidental by no means that the first steps of the formal analysis of truth (the concepts of satisfaction and definability) appeared in a paper (see Tarski 1931) in which some problems of descriptive topology were studied—a domain in which the transfinite plays an important role.

(DG6) Robert Vaught (see Vaught 1974, p. 161) reports Tarski’s dissatisfaction—expressed in his seminars in 1926–1928—that the concept of satisfaction was not properly defined. This explains a point raised by Georg Kreisel (see Kreisel 1987, p. 122):According to Andrzej Mostowski , in a conversation in Tarski’s presence, the latter and his students had no confidence in Gödel’s paper [Gödel 1930 – J. W.] when they saw the relevant issue of the Mhfte [an abbreviation for the journal in which the paper appeared – J. W.] in Warsaw. Why? Gödel had not formally defined validity ! Anybody who is surprised by this knows ipso facto that he simply had no feeling for the subject.”

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