Climax-realization below δ¹₂

Characterize all ordinals α with 0 < α < δ¹₂ such that there exists a recursive predilator D that is not a dilator (a pseudodilator) whose climax Clim(D), defined as the least ordinal at which D(α) becomes ill-founded, satisfies Clim(D) = α.

Background

The paper introduces pseudodilators (recursive predilators that are not dilators) and their climax, Clim(D), the least ordinal where D(α) is ill-founded. It is shown that for Σ¹₂-definable pseudodilators, Clim(D) < δ¹₂ and that δ¹₂ is the supremum of Clim(D) for recursive pseudodilators.

Known obstructions are identified: if α is parameter-free Σ¹₁-reflecting and either admissible or a limit of admissibles, then no recursive pseudodilator has climax α. The open problem asks for a complete characterization of which ordinals below δ¹₂ can occur as climaxes of recursive pseudodilators.