 The Cantor-Bendixson theorem shows that there is no interpolant in Γ in the case where Γ is the pointclass of closed sets, thus verifying this version of CH. This was improved by Suslin who showed that this version of CH holds for Γ where Γ is the class of Σ̰11 sets. One cannot go much further within ZFC—to prove stronger versions one must bring in stronger assumptions. It turns out that axioms of definable determinacy and large cardinal axioms achieve this. For example, results of Kechris and Martin show that if Δ̰1n-determinacy holds then this version of CH holds for the pointclass of Σ̰1n+1 sets. Going further, if one assumes ADL(ℝ) then this version of CH holds for all sets of real numbers appearing in L(ℝ). Since these hypotheses follow from large cardinal axioms one also has that stronger and stronger large cardinal assumptions secure stronger and stronger versions of this version of the effective continuum hypothesis. Indeed large cardinal axioms imply that this version of CH holds for all sets of reals in the definability hierarchy we are considering; more precisely, if there is a proper class of Woodin cardinals then this version of CH holds for all universally Baire sets of reals. 