Deductive equivalence of IΔ0 + (Σ_n) and BΣ_n^-

Determine whether, for each natural number n ≥ 0, the theory IΔ0 augmented with the parameter-free collection scheme (Σ_n) is deductively equivalent to the theory BΣ_n^- (as defined in Kaye, Paris, and Dimitracopoulos), i.e., ascertain whether IΔ0 + (Σ_n) and BΣ_n^- prove exactly the same sentences.

Background

The paper reviews multiple variants of collection schemes over PA^- and their relationships with induction schemes. In this context, the authors discuss parameter-free versions studied by Kaye, Paris, and Dimitracopoulos and by Cordón-Franco et al. Theories formed by adding these schemes to IΔ0 are central in the model-theoretic analysis the paper undertakes.

Within this landscape, two closely related parameter-free bounding theories are compared: IΔ0 + (Σn) and BΣ_n^- (introduced in prior literature). While several implications between such theories are known—e.g., BΣ{n+1}^- proving IΣ_n—the exact deductive equivalence between IΔ0 + (Σ_n) and BΣ_n^- has not been resolved. The authors explicitly note this as unknown, citing prior references that formulate it as an open problem in the literature.