Harrington's results on arithmetical singletons

TL;DR

This paper presents new proofs of two theorems by Leo Harrington regarding the existence of arithmetical singletons with specific properties, using an oracle construction instead of traditional injury methods.

Contribution

It provides alternative proofs for Harrington's theorems on arithmetical singletons and ranked points, avoiding injury methods by employing an oracle construction.

Findings

Existence of arithmetical singletons that are arithmetically incomparable

Existence of a ranked point that is not an arithmetical singleton

New proof techniques using oracle construction

Abstract

We exposit two previously unpublished theorems of Leo Harrington. The first theorem says that there exist arithmetical singletons which are arithmetically incomparable. The second theorem says that there exists a ranked point which is not an arithmetical singleton. Unlike Harrington's proofs of these theorems, our proofs do not use the finite- or infinite-injury priority method. Instead they use an oracle construction adapted from the standard proof of the Friedberg Jump Theorem.