On n-Tardy Sets

TL;DR

This paper advances the understanding of n-tardy sets in computability theory by establishing the existence of specific n-tardy sets with unique properties and answering open questions from prior research.

Contribution

It constructs examples of n-tardy sets with particular properties and resolves open questions about the structure and hierarchy of n-tardy sets.

Findings

Existence of a 3-tardy set not computed by any 2-tardy set.

Existence of properties Q_n(A) characterizing properly n-tardy sets.

Resolved all open questions posed by Harrington and Soare regarding n-tardy sets.

Abstract

Harrington and Soare introduced the notion of an n-tardy set. They showed that there is a nonempty $\mathcal{E}$ property Q(A) such that if Q(A) then A is 2-tardy. Since they also showed no 2-tardy set is complete, Harrington and Soare showed that there exists an orbit of computably enumerable sets such that every set in that orbit is incomplete. Our study of n-tardy sets takes off from where Harrington and Soare left off. We answer all the open questions asked by Harrington and Soare about n-tardy sets. We show there is a 3-tardy set A that is not computed by any 2-tardy set B. We also show that there are nonempty $\mathcal{E}$ properties $Q_n(A)$ such that if $Q_n(A)$ then A is properly n-tardy.