Scott spectral gaps for trees are bounded
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866917283630153728 |
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| author | Harrison-Trainor, Matthew Kim, J. Thomas |
| author_facet | Harrison-Trainor, Matthew Kim, J. Thomas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_07166 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Scott spectral gaps for trees are bounded Harrison-Trainor, Matthew Kim, J. Thomas Logic 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. |
| title | Scott spectral gaps for trees are bounded |
| topic | Logic |
| url | https://arxiv.org/abs/2602.07166 |