Streamlined subrecursive degree theory

This paper is divided into two parts. In Part I, we investigate the structure of honest elementary degrees, that is, the degree structure induced on the honest functions by the reducibility relation “being (Kalmar) elementary in”. In Part II, we generalise the degree theory found in Part I. We introduce the reducibility relation “being α-elementary in”, where α is an ordinal ≤ϵ0, and investigate the structure of honest α-elementary degrees. Towards the end of the paper, we discuss relations between our degree theory and provability in Peano Arithmetic.