The $\omega$-Vaught's Conjecture

TL;DR

The paper introduces the $$-Vaught's conjecture, a stronger form of the infinitary Vaught's conjecture, and provides evidence supporting it for linear orders, suggesting a new approach to the original conjecture.

Contribution

It defines the $$-Vaught's conjecture, establishes an equivalent condition, and proves it holds for all infinitary sentences with models as linear orders.

Findings

The $$-Vaught's conjecture is equivalent to a new structural condition.

All infinitary sentences with models as linear orders satisfy the $$-Vaught's conjecture.

Abstract

We introduce the $\omega$-Vaught's conjecture, a strengthening of the infinitary Vaught's conjecture. We believe that if one were to prove the infinitary Vaught's conjecture in a structural way without using techniques from higher recursion theory, then the proof would probably be a proof of the $\omega$-Vaught's conjecture. We show the existence of an equivalent condition to the $\omega$-Vaught's conjecture and use this tool to show that all infinitary sentences whose models are linear orders satisfy the $\omega$-Vaught's conjecture.