2 results
ON THE C.E. DEGREES REALIZABLE IN
$\Pi ^0_1$ CLASSES
- Part of
-
- Journal:
- The Journal of Symbolic Logic / Volume 89 / Issue 3 / September 2024
- Published online by Cambridge University Press:
- 24 April 2023, pp. 1370-1395
- Print publication:
- September 2024
-
- Article
-
- You have access
- Open access
- HTML
- Export citation
-
We study for each computably bounded
$\Pi ^0_1$ class P the set of degrees of c.e. paths in P. We show, amongst other results, that for every c.e. degree a there is a perfect 
$\Pi ^0_1$ class where all c.e. members have degree a. We also show that every 
$\Pi ^0_1$ set of c.e. indices is realized in some perfect 
$\Pi ^0_1$ class, and classify the sets of c.e. degrees which can be realized in some 
$\Pi ^0_1$ class as exactly those with a computable representation.
CUPPING AND JUMP CLASSES IN THE COMPUTABLY ENUMERABLE DEGREES
- Part of
-
- Journal:
- The Journal of Symbolic Logic / Volume 85 / Issue 4 / December 2020
- Published online by Cambridge University Press:
- 30 October 2020, pp. 1499-1545
- Print publication:
- December 2020
-
- Article
- Export citation
-
We show that there is a cuppable c.e. degree, all of whose cupping partners are high. In particular, not all cuppable degrees are
${\operatorname {\mathrm {low}}}_3$-cuppable, or indeed 
${\operatorname {\mathrm {low}}}_n$ cuppable for any n, refuting a conjecture by Li. On the other hand, we show that one cannot improve highness to superhighness. We also show that the 
${\operatorname {\mathrm {low}}}_2$-cuppable degrees coincide with the array computable-cuppable degrees, giving a full understanding of the latter class.
