A Feiner Look at the Intermediate Degrees

We say that a set \(S\) is \(\Delta^0_{(n)}(X)\) if membership of \(n\) in \(S\) is a \(\Delta^0_{n}(X)\) question, uniformly in \(n\). A set \(X\) is low for \(\Delta\)-Feiner if every set \(S\) that is \(\Delta^0_{(n)}(X)\) is also \(\Delta^0_{(n)}(\emptyset)\). It is easy to see that every low\(_n\) set is low for \(\Delta\)-Feiner, but we show that the converse is not true by constructing an intermediate c.e. set that is low for \(\Delta\)-Feiner. We also study variations on this notion, such as the sets that are \(\Delta^0_{(bn+a)}(X)\), \(\Sigma^0_{(bn+a)}(X)\), or \(\Pi^0_{(bn+a)}(X)\), and the sets that are low, intermediate, and high for these classes. In doing so, we obtain a result on the computability of Boolean algebras, namely that there is a Boolean algebra of intermediate c.e. degree with no computable copy.