A splitting theorem for simple Π11 Sets

Abstract

As was first mentioned in [3, §5], if A is any – set, A is the union of two disjoint – sets B(0), B(1). In metarecursion theory this is proven as follows. Let ƒ be a one-to-one metarecursive function whose range is A, let R be an unbounded metarecursive set whose complement is also unbounded, and set B(0) = f(R), B(1) = f(). The corresponding fact of ordinary recursion theory, namely that any r.e. but not recursive set can be split into two other such sets, was proved by Friedberg [2, Theorem 1], using a clever priority argument. Sacks [7, Corollary 2] then showed that any r.e. but not recursive set is the union of two disjoint r.e. sets neither of which was recursive in the other, a much stronger result. In this paper we attempt to prove the analogous result for – sets A, but succeed only in the case A is simple; i.e., the complement of A contains no infinite subset. As a corollary we show the metadegrees are dense, a fact already announced by Sacks [8, Corollary 1], but only proven by him for nonzero metadegrees.