Beyond strong jump traceability
Keng Meng Ng · Proceedings of the London Mathematical Society · 2010
Strong jump traceability has been studied by various authors. In this paper we study a variant of strong jump traceability by looking at a partial relativization of strong jump traceability. We discover a new subclass ℋ of the computably enumerable (c.e.) K-trivials with some interesting properties. These sets are computationally very weak, but yet contain a cuppable member. Surprisingly they cannot be constructed directly using cost functions, and is the first known example of a subclass of the K-trivials which does not contain any promptly simple member. Furthermore, there is a single c.e. set which caps every member of ℋ, demonstrating that they are in fact very far away from being promptly simple.