##
A Feiner Look at the Intermediate Degrees

Status: submitted

Availability: preprint

**Abstract.** We say that a set *S* is
Δ^{0}_{(n)}(*X*) if
membership of *n* in *S* is a
Δ^{0}_{n}(*X*) question,
uniformly in *n*. A set *X* is *low for Δ-Feiner*
if every set *S* that is
Δ^{0}_{(n)}(*X*)
is also Δ^{0}_{(n)}(∅) . It is
easy to see
that every low_{n} set is low for Δ-Feiner, but we
show that the
converse is not true by constructing an intermediate c.e. set that is low
for Δ-Feiner. We also study variations on this notion, such as the
sets that are
Δ^{0}_{(bn+a)}(*X*),
Σ^{0}_{(bn+a)}(*X*), or
Π^{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.

drh@math.uchicago.edu