Abstract

This article is essentially a part of my thesis for the degree DrSc (Doctor of Sciences). Therefore it mainly surveys my articles [41, 42, 43, 28, 29, 44, 22], and it is structured according to the requirements for such theses. I made only minor changes in the original text and added a few further references. Since Godel's main achievement concerns the problem of consistency and some of the problems that I am going to describe had been considered by him, I think that it is appropriate to publish this article in Godel Society. 1 Historical remarks The question that we are going to consider in is interesting per se and is related to some more practical questions, especially in complexity theory, but the original motivation for it comes from foundational studies. Among the variety of streams in foundations of mathematics, the one which had the biggest influence and which very much determined later development of mathematical logic was Hilbert's<F1

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