@ North-Holland Publishing Cornpany 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. A few years ago S.D. Friedman and G.E. Sacks [1] started a new chapter in generalized recursion theory: p-recursion theory. So far recursion theory was studied only on those initial segments L- of the constructible hierarchy where a is admissible. In p-recursion theory one considers initial segments Lu lor any limi