Cam anyone solve this? (ZFC + PD) Let T be a Lω1,ω-sentence with uncountably many countable models. The following are equivalent: (V1) T is a counterexample to Vaught’s conjecture. (V2) T satisfies hyperarithmetic-is-recursive on a cone. (V3) There exists an oracle relative to which, {Sp(A) : A |= T} = {{X ∈ 2 ω : ω X 1 ≥ α} : α ∈ ω1}.
