Scott spectral gaps for trees are bounded

TL;DR

This paper demonstrates that for certain classes of trees, there exist models with bounded Scott rank related to the class’s definability, providing a new proof of a known fact without relying on Vaught's conjecture.

Contribution

It establishes bounds on Scott ranks for trees within definable classes, offering a novel proof that trees are not Borel complete.

Findings

Existence of trees with Scott rank at most α+2 for classes definable by a Π_α sentence

New proof that trees are not faithfully Borel complete

Bounded Scott ranks linked to class definability

Abstract

Given a Borel class of trees, we show that there is a tree in that class whose Scott sentence is not too much more complicated than the definition of the class. In particular, if the class is definable by a $\Pi_\alpha$ sentence, then there is a model of Scott rank at most $\alpha + 2$. This gives another proof-and one that does not require first proving Vaught's conjecture for trees-of the fact that trees are not faithfully Borel complete.