Admissibility of s¹₂(T)

Determine whether, for every Σ¹₂-sound recursively enumerable extension T of ACA₀, the Σ¹₂ proof-theoretic ordinal s¹₂(T) is always either an admissible ordinal or a limit of admissible ordinals.

Background

The paper defines the Σ¹₂ proof-theoretic ordinal s¹₂(T) as the supremum of climaxes of recursive pseudodilators whose non-dilatorhood is provable in T. Several values are computed for natural theories (e.g., s¹₂(ACA₀) = ω¹CK, s¹₂(Π¹₁–CA₀) = ω^CK_ω).

Motivated by these examples and structural results about δ¹₂, the author raises the question of whether s¹₂(T) must always be an admissible or a limit of admissibles for any Σ¹₂-sound r.e. extension of ACA₀.