Degrees of functionals

Abstract

In this paper we will discuss some problems of degree-theoretic nature in connection with recursion in normal objects of higher types. Harrington [2] and Loewenthal [6] have proved some results concerning Post's problem and the Minimal Pair Problem, using recursion modulo subindividuals. Our degrees will be those obtained from Kleene-recursion modulo individuals. To solve our problems we then have to put some extra strength to ZFC. We will first assume V = L, and then we restrict ourselves to the situation of a recursive well-ordering and Martin's axiom. We assume familiarity with recursion theory in higher types as presented in Kleene [3]. Further backround is found in Harrington [2], Moldestad [9] and Normann [11]. We will survey the parts of these papers that we need. In Section 1 we give the general background for the arguments used later. In Section 2 we prove some lemmas assuming V = L. In section 3, assuming V = L we solve Post's problem and another problem using the finite injury method. We will thereby describe some of the methods needed for the more complex priority argument of Section 4 where we give a solution of the minimal pair problem for extended r.e. degress of functionals. In Section 5 we will see that if Martin's Axiom holds and we have a minimal well-ordering of tp (1) recursive in 3 E, we may use the same sort of arguments as in parts 3 and 4.