Kavli Affiliate: Cheng Peng
| First 5 Authors: Yiqun Liu, Yiqun Liu, , ,
| Summary:
A Turing degree is d.c.e. if it contains a set that is the difference of two
c.e. sets. A d.c.e. degree $mathbfd$ is isolated by a c.e. degree
$mathbfa<mathbfd$ if all c.e. degrees that are below $mathbfd$ are
also below $mathbfa$; $mathbfd$ is isolated from above by a c.e. degree
$mathbfa>mathbfd$ if all c.e. degrees that are above $mathbfd$ are
also above $mathbfa$. In this paper, we study the inductive strength of both
isolated and upper isolated d.c.e. degrees from the point of view of reverse
recursion theory. We show that (1) $P^- + BSigma_1 + textExp vdash
ISigma_1 leftrightarrow$ There is an isolated proper d.c.e. degree below
$mathbf0’$; (2) $P^- + BSigma_1 + textExp vdash ISigma_1
leftrightarrow$ There is an upper isolated proper d.c.e. degree below
$mathbf0’$.
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