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Alfred Tarski's work on general metamathematics

  • W. J. Blok (a1) and Don Pigozzi (a2)

In this essay we discuss Tarski's work on what he called the methodology of the deductive sciences, or more briefly, borrowing the terminology of Hilbert, metamathematics, The clearest statement of Tarski's views on this subject can be found in his textbook Introduction to logic [41m].1 Here he describes the tasks of metamathematics as “the detailed analysis and critical evaluation of the fundamental principles that are applied in the construction of logic and mathematics”. He goes on to describe what these fundamental principles are: All the expressions of the discipline under consideration must be defined in terms of a small group of primitive expressions that seem immediately understandable. Furthermore, only those statements of the discipline are accepted as valid that can be deduced by precisely defined and universally accepted means from a small set of axioms whose validity seems evident. The method of constructing a discipline in strict accordance with these principles is known as the deductive method, and the disciplines constructed in this manner are called deductive systems. Since contemporary mathematical logic is one of those disciplines that are subject to these principles, it itself is a deductive science. Tarski then goes on to say:

“The view has become more and more common that the deductive method is the only essential feature by means of which the mathematical disciplines can be distinguished from all other sciences; not only is every mathematical discipline a deductive theory, but also, conversely, every deductive theory is a mathematical discipline”.

This identification of mathematics with the deductive sciences is in our view one of the distinctive aspects of Tarski's work. Another characteristic feature is his broad view of what constitutes the domain of metamathematical investigations. A clue to this aspect of his work can also be found in Chapter 6 of Introduction to logic. After a discussion of the notions of completeness and consistency, he remarks that the investigations concerning these topics were among the most important factors contributing to a considerable extension of the domain of methodological studies, and caused even a fundamental change in the whole character of the methodology of deductive sciences.

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