Remember that an n-REA set is a set of the form $A\_0 \oplus A\_1 \oplus \cdots \oplus A\_n$ with $A\_n$ relatively r.e. in $A\_m, m<n$ (so $A\_0$ is r.e.) and that a degree is PA just if it computes a path through every infinite computable tree.


By Arslanov's completeness criterion no incomplete r.e. set can be of PA degree. While pursuing another problem I'm pretty sure I constructed a low 2-REA set of PA degree but before I bother to include that writeup in some paper (it's not the most riveting proof but seems worth mentioning somewhere) or even spend a bunch of time double checking it I wanted to see if it's known.


So anyone know if this result (or refutation) is already in the literature or is trivial to derive from it (e.g., clever use of relativized jump inversion)?



2-REA PA degrees
Theorem 5.1 in [Recursively Enumerable Sets Modulo Iterated Jumps and Extensions of Arslanov's Completeness Criterion](https://www.jstor.org/stable/2274816?seq=1) states that Arslanov's completeness criterion holds for $n$-REA sets. The paper is from 1989 and by Jockusch, Lerman, Soare, and Solovay.


