-Recursion Theory and Ordinal Com- putability

Motivated by a talk of S. D. Friedman at BIWOC we show that the -recur-sive and -recursively enumerable sets of G. Sacks ' s -recursion theory are exactly those sets that are recursive and recursively enumerable by an ordinal Turing machines with tapes of length and time bound . 1 Introduction. -Recursion theory is a branch of higher recursion theory that was developed by G. Sacks and his school between 1 965 and 1 980. Sacks gave the following char-acterization [4]: -recursion theory lifts classical recursion theory from! to an arbi