Abstract

This chapter discusses general proof theory. The name proof theory was originally given by Hilbert to a constructive study of proofs with certain specific aims. By such a study, the consistency of mathematics is established or, more generally, a reduction of mathematics to a certain constructive part is obtained. Hence, the study of proofs was only a tool to obtain this reduction, and thus could not use principles that were more advanced than the principles contained in the constructive part of mathematics to which all mathematics was to be reduced. Some of the more important results in reductive proof theory, in particular Gentzen's well-known results, were indeed such by-products. They were obtained from insights about the general structure of proofs, which insights are in their own right at least as interesting as their applications in reductive proof theory. It is tried to formulate this method in more general terms, in the chapter, by not tying the notion of validity of inferences to any particular deductive system but defining it for inferences in general. Results mentioned above may then be obtained as special cases of this method. The work is thus directed towards a foundation of general proof theory, but it should be noted that there are many open problems in this connection and the work has a tentative character.

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