Eventual Π¹₂ dilator value versus Σ¹₂-reflection rank

Ascertain whether, for Π¹₂-sound recursively enumerable extensions S and T of ACA₀, the inequality rank^{Σ¹₂}_{Π¹₂}(S) ≤ rank^{Σ¹₂}_{Π¹₂}(T) holds if and only if the eventual values of their proof-theoretic dilators satisfy |S|_{Π¹₂}(δ¹₂) ≤ |T|_{Π¹₂}(δ¹₂).

Background

The paper shows nonlinearity phenomena for Π¹₂-consequences modulo Σ¹₂-oracles, but develops a robust ranking via the relation S ≺^{{Σ¹₂}_{Π¹₂}} T defined by T ⊢^{Σ¹₂} RFNΠ¹₂. It proposes that the eventual value of the proof-theoretic dilator |T|_{Π¹₂} at δ¹₂ may capture this rank.

The conjecture asks for an equivalence between comparisons of the Σ¹₂-modulo Π¹₂ reflection rank and comparisons of the eventual dilator values at δ¹₂, thus providing an ordinal characteristic for the Π¹₂ setting analogous to results established for Σ¹₂ in the paper.

References

We conjecture the eventual value of a proof-theoretic dilator |T|{\Pi^{1_2}(\delta^1_2)} should have this role: Let rank^{{\Sigma^1_2}}{\Pi^1_2}(T) be the \prec^{{\Sigma^{1_2}_{\Pi^1_2}-rank}} of T for \Pi^1_2-sound r.e. extension T of ACA_0. Then does the following hold? For two \Pi^1_2-sound r.e. extensions S, T of ACA_0, rank^{{\Sigma^{1_2}_{\Pi^1_2}(S)}} \le rank^{{\Sigma^{1_2}_{\Pi^1_2}(T)}} iff |S|{\Pi^{1_2}(\delta^1_2)} \le |T|{\Pi^{1_2}(\delta^1_2).}

— The behavior of higher proof theory I: Case $Σ^1_2$ (2406.03801 - Jeon, 2024) in Section 7: A glimpse to Case Π¹₂

Eventual Π¹₂ dilator value versus Σ¹₂-reflection rank

Background

References

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