The Solecki Dichotomy and the Posner-Robinson Theorem are Almost Equivalent

TL;DR

This paper reveals a deep connection between the Solecki dichotomy in descriptive set theory and the Posner-Robinson theorem in computability theory, showing they are almost equivalent under certain conditions.

Contribution

It establishes a formal relationship between the two theorems by formulating weakened versions and demonstrating their equivalence with determinacy principles.

Findings

Weak versions of the theorems are equivalent under determinacy.

The relationship extends to higher levels of the hierarchy.

Each theorem can be used to prove the other with the right assumptions.

Abstract

The Solecki dichotomy in descriptive set theory and the Posner-Robinson theorem in computability theory bear a superficial resemblance to each other and can sometimes be used to prove the same results, but do not have any obvious direct relationship. We show that in fact there is such a relationship by formulating slightly weakened versions of the two theorems and showing that, when combined with determinacy principles, each one yields a short proof of the other. This relationship also holds for generalizations of the Solecki dichotomy and the Posner-Robinson theorem to higher levels of the Borel/hyperarithmetic hierarchy.