 But Gödel’s hope that large cardinal axioms could solve the continuum problem turned out to be unfounded. The continuum hypothesis is independent of ZFC even in the context of large cardinal axioms. Nevertheless, large cardinal principles have manage to settle restricted versions of the continuum hypothesis (in the affirmative). The existence of so-called Woodin cardinals ensures that sets definable in analysis are either countable or the size of the continuum. Thus the definable continuum problem is settled.