CH: every infinite subset of \mathbb{R} is either countable or equinumerous with \mathbb{R}. Asked as posed, the problem has no truth value decidable from the standard axioms: ZFC ⊬ CH and ZFC ⊢ ¬CH is equally impossible — CH is independent of ZFC.
Cantor spent his life on it; Hilbert placed it FIRST on his 1900 list. Gödel (1938) built the constructible universe L ⊨ CH; Paul Cohen (1963) invented forcing to build models with ¬CH, winning the Fields Medal for it — the only problem on Hilbert's list resolved by proving no answer exists within the accepted framework.
Independence established — Gödel 1938/1940, Cohen 1963. The resolution redefined mathematical ontology: pluralism about universes of sets (the multiverse view) descends directly from it. Woodin's Ω-logic programme and the inner-model programme continue the hunt for 'the' right axioms; whether the multiverse or a future axiom settles CH in the eyes of practice remains the deepest meta-question in foundations. Not solved in the naive sense — solved in the deepest sense available: we learned exactly what kind of question it is.