Abstract
Publisher Summary This chapter describes a fine structure on hierarchies generated by reflection formulas over primitive recursive arithmetic (PRA) and its extensions. The comparison of the hierarchies nα or the iterated reflection formulas, involving other proof theoretical concepts and the application of the fine structure theorem, yields a lot of results that are much easier to obtain by the fine structure than by their original proofs. This method is a useful tool for dealing with problems concerning consistency and reflection principles, transfinite induction, proofs of restricted complexity, and some other concepts. The usefulness of iterated reflection formulas and the fine structure become evident by comparing the Cn to other proof theoretical concepts and by showing some applications. The chapter discusses induction and iterated reflection formulas, transfinite induction and iterated reflection formulas, fine structure relations for other theories than PRA, and k-consistency and iterated reflection formulas.
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