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A counterexample to Vaught's Conjecture for $L_{\omega_1,\omega}$

Contributed by Nathanael L. Ackerman, Cameron Freer, Robin W. Knight

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.

Primarily human-written textHuman understanding: all partsKnight systemsVaught’s conjectureinfinitary logicnumber of models

Oct 11, 2026PDF ↗View work →

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A counterexample to Vaught's Conjecture for $L_{\omega_1,\omega}$

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A counterexample to Vaught's Conjecture for Lω1,ωL_{\omega_1,\omega}Lω1​,ω​

A counterexample to Vaught's Conjecture for $L_{\omega_1,\omega}$