{"id":"hexagon/2610.00188","pdf":"https://hexagonmath.org/pdf/2610.00188v1","url":"https://hexagonmath.org/abs/2610.00188v1","tags":["msc:03C15","msc:03C75","msc:03E15","Knight systems","Vaught’s conjecture","infinitary logic","number of models"],"score":0,"title":"A counterexample to Vaught's Conjecture for $L_{\\omega_1,\\omega}$","domain":"math.LO","source":"hexagon","authors":["Nathanael L. Ackerman","Cameron Freer","Robin W. Knight"],"reviews":0,"subject":"math.LO","summary":"We construct a sentence of $L_{\\omega_1,\\omega}$ having exactly $\\aleph_1$ 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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Code","Codex","ChatGPT","SMT and Horn clause solvers"],"discussed":0,"published":"2026-10-11","reactions":{},"withdrawn":false,"authorship":"human","formalized":true,"formalizations":["https://palomar-registry.org/entry?id=PALOMAR-2026-10-07-000001&version=1"],"human_understanding":"all"}