ON c - AND P - RECURSMLY ENUMERABLE DEGREES

Several problems in recursion theory on admissible o¡dinals (a -recursion theory) and recursion theory on inadmissible ordinals (B -recursion theory) are studied. Fruitful interactions betweenboth theories are stressed. In the fr¡stpart the admissible collapse is used in order to characterizefor some inadmissible B the structure of all p -recursively enumerable degrees asanaccumulationof structures of 2[-recursively enumerable degrees for many admissible structures 2[. Thus problems about the B -recursively enumerable degrees can be solved by considering "locally" the analogous problem in an admissible 2[ (where results of a -recursion theory apply). In the second part p -recursion theory is used as a tool in infinite injurypriority constructions for some particularly interesting c (e.g. ,1"). New effects canbeobserved since some structu¡e of the inadmissibleworld above O'isprojected into the a-recursivelyenumerable degrees byinvertingthejump. The gained understandingof thejump of a-recursively enumerable degrees makes it possible to solve some open problems.