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PreprintCC BY 4.0arXiv

A counterexample to Vaught's Conjecture for Lω1,ω

  • Nathanael L. Ackerman
  • Cameron E. Freer
  • Robin W. Knight
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Abstract

We construct a sentence of Lω1,ω having exactly ℵ1 many countable models up to isomorphism, but no perfect set of non-isomorphic models. This disproves Vaught's conjecture for infinitary logic. The construction follows a framework devised by Robin Knight, and the proof has been formally verified in Lean.

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Cite

BibTeXRIS
BibTeX
@misc{ackerman2026counterexample,
  title = {{A counterexample to Vaught's Conjecture for $L_ω_1,ω$}},
  author = {Nathanael L. Ackerman and Cameron E. Freer and Robin W. Knight},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15217},
  eprint = {2610.15217},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15217},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A counterexample to Vaught's Conjecture for L_ω_1,ω
AU  - Nathanael L. Ackerman
AU  - Cameron E. Freer
AU  - Robin W. Knight
PY  - 2026
DA  - 2026/10/10
PB  - arXiv
DO  - 10.48550/arxiv.2610.15217
UR  - https://arxiv.org/abs/2610.15217
AB  - We construct a sentence of L_ω_1,ω having exactly ℵ_1 many countable models up to isomorphism, but no perfect set of non-isomorphic models. This disproves Vaught's conjecture for infinitary logic. The construction follows a framework devised by Robin Knight, and the proof has been formally verified in Lean.
ER  -