There is no plus-capping degree

Answering a question of Per Lindstrom, we show that there is no \plus-capping" degree, i.e. that for any incomplete r.e. degree w, there is an incomplete r.e. degree a > w such that there is no r.e. degree v > w with a \ v = w. 1. The Theorem. In [2], Feferman de nes relative interpretability for extensions of Peano arithmetic. The induced structure D T of degrees of interpretability turns out to be a lattice (cf. [4]). Lindstrom [5] proved a number of algebraic facts about this lattice. He recently raised the question whether one of these, namely 8w < 1 9a (w < a < 1 ^ 8v > w:(a ^ v = w)); also holds in the upper semilattice of r.e. degrees. This question turns out to be of independent interest to classical recursion theorists as they try to better understand the algebraic structure of the r.e. degrees. The purpose of this paper is to answer Lindstrom's question positively by proving the following