PreprintCC BY 4.0arXiv
The Type ω-Vaught's Conjecture for ω-stable Theories
Abstract
We propose a new variant of Vaught's conjecture, based on Martin's (model-theoretic) conjecture and the ω-Vaught's conjecture of Gonzalez and Montalbán. For ω-stable theories, we prove the conjecture by analyzing in detail Bouscaren's and Shelah-Harrington-Makkai's proofs of Martin's conjecture and Vaught's conjecture, respectively, for ω-stable theories. In particular, when there are countably many countable models, we improve the required number of quantifiers for a Scott sentence from ω+ω to ω+5, and we obtain a similar bound of ω+6 for the uncountable case.
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BibTeX
@misc{zhu2026type,
title = {{The Type $ω$-Vaught's Conjecture for $ω$-stable Theories}},
author = {Hongyu Zhu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09314},
eprint = {2610.09314},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09314},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Type ω-Vaught's Conjecture for ω-stable Theories
AU - Hongyu Zhu
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09314
UR - https://arxiv.org/abs/2610.09314
AB - We propose a new variant of Vaught's conjecture, based on Martin's (model-theoretic) conjecture and the ω-Vaught's conjecture of Gonzalez and Montalbán. For ω-stable theories, we prove the conjecture by analyzing in detail Bouscaren's and Shelah-Harrington-Makkai's proofs of Martin's conjecture and Vaught's conjecture, respectively, for ω-stable theories. In particular, when there are countably many countable models, we improve the required number of quantifiers for a Scott sentence from ω+ω to ω+5, and we obtain a similar bound of ω+6 for the uncountable case.
ER -