Extending Properly n-REA Sets

TL;DR

This paper disproves the hypothesis that every properly n-REA set can be extended to a properly n+1-REA set, by providing a counterexample for n=3.

Contribution

It demonstrates that the natural extension of previous results does not hold universally, showing limitations in the structure of properly n-REA sets.

Findings

Counterexample for n=3-REA sets

Disproof of the extension hypothesis for properly n-REA sets

Limits on the extendability of properly n-REA sets

Abstract

In [5] Soare and Stob prove that if $A$ is an r.e. set which isn't computable then there is a set of the form $A \oplus W^A_e$ which isn't of r.e. Turing degree. If we define a properly $n+1$-REA set to be an $n+1$-REA set which isn't Turing equivalent to any $n$-REA set this result shows that every properly 1-REA set can be extended to a properly 2-REA set. This result was extended in [1] by Cholak and Hinman who proved that every 2-REA set can be extended to a properly 3-REA set. This leads naturally to the hypothesis that every properly $n$-REA set can be extended to a properly $n+1$-REA set. In this paper, we show this hypothesis is false and that there is a properly $3$-REA set which can't be extended to a properly $4$-REA set.