A note on Hjorth's oscillation theorem

TL;DR

This paper reformulates Hjorth's oscillation theorem within continuous logic, providing a simpler proof and clarifying its connection to first-order logic.

Contribution

It introduces a new, simplified proof of Hjorth's oscillation theorem in the context of continuous logic, enhancing understanding of its relation to first-order logic.

Findings

Simplified proof of Hjorth's oscillation theorem in continuous logic

Clarification of the relation between Hjorth's theorem and first-order logic

Enhanced conceptual understanding of oscillation phenomena

Abstract

We reformulate, in the context of continuous logic, an oscillation theorem originally proved by G. Hjorth. We give a proof of the theorem in that setting which is similar to, but simpler than, Hjorth's original one. The point of view presented here clarifies the relation between Hjorth's theorem and first-order logic.