Extending properly n - REA sets1

In 1982, Soare and Stob proved that if A is an r.e. set which isn’t computable then there is a set of the form A ⊕ W 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 by Cholak and Hinman in 1994. They 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. Here 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. Moreover we show this set A can be Δ 2 0 .