Vaught's Conjecture

Claim: every complete first-order theory T in a countable language has either ≤ ℵ₀ non-isomorphic countable models or exactly \(2^{ℵ₀}\) such models. Never some intermediate cardinal.

Vaught proved the two-cardinal version (never strictly between ℵ₀ and continuum) for theories with finitely many relations in 1961. Morley's 1970 categoricity theorem showed that if T is categorical in SOME uncountable power, it has few countable models. Shelah's classification programme (main gap theorem) resolved the superstable case. The general case remains the field's most-wanted theorem after the Main Gap itself.