Type ω-Vaught’s conjecture for ω-stable theories

The conjecture asks for a Vaught-type classification of the countable models of every countable complete $\omega$ -stable theory $T$ , based on the model-theoretic Martin conjecture and the $\omega$ -Vaught conjecture of González and Montalbán. The supplied sources do not state the proposed type- $\omega$ classification in a sufficiently precise quantified form to reconstruct its full formal statement. They do state that the conjecture predicts the relevant dichotomy for countable models and that, in the case of countably many countable models, the associated Scott sentences can be taken with quantifier complexity $\omega+5$ , with an analogous bound $\omega+6$ in the uncountable case.

References

Primary source

arXiv

Additional references

The Type ω-Vaught's Conjecture for ω-stable Theories — arXiv — Hongyu Zhu

descriptive-set-theory model-theory

Posted 4 hours ago · edited 3 hours ago

Progress summary

Refreshed 3 hours ago

Claimed progress

A new unrefereed preprint claims a positive result for the restricted setting, but the full classification question remains open.

The problem asks for a Vaught-type classification of countable models of $\omega$ -stable theories. The latest preprint claims progress through bounds on the complexity of the relevant Scott sentences.

Known results

Shelah, Harrington, and Makkai (1983–1984) proved ordinary Vaught’s conjecture for countable complete $\omega$ -stable theories: there are either countably many or continuum many non-isomorphic countable models.
Their work also characterized the theories with fewer than continuum many countable models and established structural results for those models.
Later work showed Borel completeness in important $\omega$ -stable cases, including ENI-DOP and ENI-NDOP with ENI-depth; a full characterization remains open.

2026 preprint: claimed positive regime

Hongyu Zhu’s preprint analyzes the $\omega$ -stable cases of Martin’s and Vaught’s conjectures and claims Scott-sentence bounds of $\omega+5$ and $\omega+6$ . An earlier preprint established only an $(\omega\cdot 2)$ -version, so the full claim remains unverified; no independent assessment was found.

Current status (as of October 2026): Classical Vaught-type results for $\omega$ -stable theories are settled, while Zhu’s claimed stronger bounds and the full Type $\omega$ -Vaught conjecture remain unverified and open.

Sources

No solutions have been posted yet.