Scott spectral gaps for trees are bounded

Matthew Harrison-Trainor; J. Thomas Kim

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arXiv · 2602.07166

Scott spectral gaps for trees are bounded

Matthew Harrison-Trainor·J. Thomas Kim

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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.

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Matthew Harrison-Trainor, J. Thomas Kim. 2026-02-06. Scott spectral gaps for trees are bounded. https://arxiv.org/abs/2602.07166

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