Extending Properly n-REA Sets
Peter A. Cholak, Peter M. Gerdes

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 is an r.e. set which isn't computable then there is a set of the form which isn't of r.e. Turing degree. If we define a properly -REA set to be an -REA set which isn't Turing equivalent to any -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 -REA set can be extended to a properly -REA set. In this paper, we show this hypothesis is false and that there is a properly -REA set which can't be extended to a properly -REA set.
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Taxonomy
TopicsRough Sets and Fuzzy Logic
