Superhighness

TL;DR

This paper explores the properties of superhigh sets in computability theory, proving they can be jump traceable but not weakly 2-random, and analyzing the class superhigh$^ riangle$, revealing its relation to noncomputable $K$-trivial sets.

Contribution

It demonstrates that superhigh sets can be jump traceable and characterizes the superhigh$^ riangle$ class in relation to $K$-trivial sets, answering open questions.

Findings

Superhigh sets can be jump traceable.

Superhigh sets cannot be weakly 2-random.

Superhigh$^ riangle$ contains some noncomputable $K$-trivial sets.

Abstract

We prove that superhigh sets can be jump traceable, answering a question of Cole and Simpson. On the other hand, we show that such sets cannot be weakly 2-random. We also study the class superhigh$^\Diamond$, and show that it contains some, but not all, of the noncomputable $K$-trivial sets.