On the Nonexistence of a Strong Minimal Pair

Abstract

Two nonzero recursively enumerable (r.e.) degrees $\mathbf{a}$ and $\mathbf{b}$ form a strong minimal pair if $\mathbf{a}\wedge \mathbf{b}=0$ and $\mathbf{b}\vee \mathbf{x}\ge \mathbf{a}$ for any nonzero r.e. degree $\mathbf{x}\le \mathbf{a}$. We prove that there is no strong minimal pair in the r.e. degrees. Our construction goes beyond the usual $0^{\prime \prime \prime }$-priority arguments and we give some evidence to show that it needs $0^{(4)}$-priority arguments.