Transitive model characterization of s¹₂(T)

Prove that for any Σ¹₂-sound extension T of Π¹₁–CA₀, the set theory ATR₀^set augmented by all Σ¹₂-consequences of T has a transitive model, and moreover that s¹₂(T) equals the minimum ordinal height of such a transitive model, i.e., s¹₂(T) = min{ ord(N) | N is a transitive model of ATR₀^set + Σ¹₂(T) }.

Background

Based on observed ties between s¹₂(T) values and the least heights of transitive models for various set theories, the author formulates a general conjecture connecting the Σ¹₂ proof-theoretic ordinal s¹₂(T) with the minimal height of transitive models of ATR₀set plus the Σ¹₂-consequences of T.

This would provide a model-theoretic characterization of s¹₂(T), extending known correspondences for specific theories and strengthening the interpretative meaning of s¹₂(T) as a robustness measure of Σ¹₂ consequences.

References

Conjecture Let T be a \Sigma1_2-sound extension of \Pi1_1\mhyphenCA_0, and \Sigma1_2(T) be the set of all \Sigma1_2-consequences of T. Then ATR_0\mathsf{set} + \Sigma1_2(T) has a transitive model, and moreover s1_2(T) = \min{N\cap \Ord\mid \text{$N$ is a transitive model of }ATR_0\mathsf{set} + \Sigma1_2(T)}.

— The behavior of higher proof theory I: Case $Σ^1_2$ (2406.03801 - Jeon, 6 Jun 2024) in Section 8: Concluding remarks (Finale)