Extending properly n - REA sets 1

Abstract

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 $\mathit{A}\oplus {\mathit{W}}_{\mathit{e}}$ which isn’t of r.e. Turing degree. If we define a properly $\mathit{n}+1$ -REA set to be an $\mathit{n}+1$ -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 $\mathit{n}+1$ -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 ${\mathrm{\Delta }}_2^0$ .