On the Nonexistence of a Strong Minimal Pair

TL;DR

This paper proves that strong minimal pairs do not exist among recursively enumerable degrees, using advanced priority arguments that likely require fourth Turing jump complexity.

Contribution

It establishes the nonexistence of strong minimal pairs in r.e. degrees, employing a novel construction that surpasses traditional priority methods.

Findings

No strong minimal pairs in r.e. degrees.

Construction requires $ extbf{0}^{(4)}$-priority arguments.

Advances understanding of the structure of r.e. degrees.

Abstract

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