A Feiner Look at the Intermediate Degrees

TL;DR

This paper explores the concept of low for -Feiner sets, constructs an intermediate c.e. set with this property, and investigates related classes, culminating in a result about the computability of Boolean algebras.

Contribution

It introduces the notion of low for -Feiner sets, constructs an intermediate c.e. set with this property, and studies variations leading to new insights in computability theory.

Findings

Constructed an intermediate c.e. set low for -Feiner.

Showed that not all low sets are low for -Feiner.

Proved the existence of a Boolean algebra of intermediate c.e. degree with no computable copy.

Abstract

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.