The 𝜔-Vaught’s conjecture

Abstract

We introduce the ω \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 ω \omega -Vaught’s conjecture. We show the existence of an equivalent condition to the ω \omega -Vaught’s conjecture and use this tool to show that all infinitary sentences whose models are linear orders satisfy the ω \omega -Vaught’s conjecture.