On α- and β-recursively enumerable degrees

Several problems in recursion theory on admissible ordinals (α-recursion theory) and recursion theory of inadmissible ordinals (β-recursion theory) are studied. Fruitful interactions between both theories are stressed. In the first part of the admissible collapse is used in order to characterize for some inadmissible β the structure of all β-recursively enumerable degrees as an accumulation of structures of $\text{U}$-recursively enumerable degrees for many admissible structures $\text{U}$. Thus problems about the β-recursively enumerable degrees can be solved by considering “locally” the analogous problem in an admissible $\text{U}$ (where results of α-recursion theory apply). In the second part β-recursion theory is used as a tool in infinite injury priority constructions for some particularly interesting α (e.g. ω₁^CK). New effects can be observed since some structure of the inadmissible world above O′ is projected into the α-recursively enumerable degrees by inverting the jump. The gained understanding of the jump of α-recursively enumerable degrees makes it possible to solve some open problems.