A {\omega}-REA Set Forming A Minimal Pair With 0'

TL;DR

The paper investigates the minimal pair relationships between various REA sets and the Turing jumps 0' and 0'', ultimately showing limitations on such pairs and constructing specific examples to demonstrate these properties.

Contribution

It proves that no { extalpha}-REA set can form a non-trivial minimal pair with 0'', and constructs a non-computable set from 0'' that forms a minimal pair with 0'.

Findings

No { extalpha}-REA set forms a minimal pair with 0''.

Constructed a non-computable set from 0'' forming a minimal pair with 0'.

Extended understanding of minimal pair relationships in computability theory.

Abstract

It is easy to see that no n-REA set can form a (non-trivial) minimal pair with 0' and only slightly more difficult to observe that no {\omega}-REA set can form a (non-trivial) minimal pair with 0". Shore has asked whether this can be improved to show that no {\omega}-REA set forms a (non-trivial) minimal pair with 0'. We show that no such improvement is possible by constructing a non-computable set C computable from 0" forming a minimal pair with 0'. We then show that no {\alpha}-REA set can form a (non-trivial) minimal pair with 0".