Scott spectral gaps for trees are bounded

TL;DR

The paper proves that for any satisfiable $\Pi_\alpha$ sentence about rooted trees, one can realize it with a tree whose Scott sentence lies at a modest level of the infinitary hierarchy, achieving a bound on the Scott rank. The core method is a forcing/Henkin construction of a generic tree with a robust property $(*)$, combined with an analysis showing that automorphism orbits become definable at level $\mathfrak{E}_{\alpha+1}$, which yields a $Π_{\alpha+2}$ Scott sentence and thus Scott rank at most $\alpha+1$ (and, in the main statement, a $Π_{\alpha+3}$ sentence giving rank at most $\alpha+2$). This establishes bounded Scott spectral gaps for trees and implies that the class of trees is not faithfully Borel complete, without requiring full Vaught's conjecture for graphs. The paper also provides a lower-bound result showing that the upper bounds are nontrivial by constructing a $\Pi_2$ theory of coloured trees with all models having a $Π_3$ but no $Σ_3$ Scott sentence.

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 $Π_α$ sentence, then there is a model of Scott rank at most $α+ 2$. 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.

Theorems & Definitions (31)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Theorem 2.2: Karp Karp
• Definition 2.3
• Lemma 2.4: Chen, Gonzalez, and Harrison-Trainor CGHT
• Lemma 2.5: Chen, Gonzalez, and Harrison-Trainor CGHT
• Lemma 3.1
• proof