Non-trivial theories in the symmetric difference between hyperarithmetic analysis and RFN^{-1}(ATR)

Identify non-trivial recursively axiomatizable L-theories that lie in the symmetric difference between the class of hyperarithmetic analysis theories (as defined by ω-models HYP(X)) and the class RFN^{-1}(ATR).

Background

The paper compares hyperarithmetic analysis (HA) and RFN{-1}(ATR), establishing several similarities and closure properties. It notes that only trivial examples are known in the symmetric difference between these two classes (e.g., Σ¹₁AC + Con(ATR)), and calls for identifying substantive, non-trivial theories that distinguish the classes.

Earlier in the development, the author highlights that despite the approximations achieved by RFN{-1}(ATR), understanding precise differences between HA and RFN{-1}(ATR) remains an open direction.

References

As previously mentioned, the following two issues remain unresolved: What non-trivial theories exist in the symmetric difference between hyperarithmetic analysis and RFN{-1}(ATR)?

— Approximation of hyperarithmetic analysis by $ω$-model reflection (2411.16338 - Hashimoto, 25 Nov 2024) in Section 6 (Open Problem)