Scott spectral gaps for trees are bounded
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.
Fields of Study & Subjects
Advanced Topology and Set Theory
Computability
Logic
AI Algorithms
Mathematical Dynamics and Fractals
Publication details
- Journal / source
- arXiv (Cornell University)
- Publication date
- 6 February 2026
- Language
- EN
- Document type
- Preprint