Higher-level Γ-Subset implications: Σn-Subset versus ∆n+1-Subset

Investigate whether there exists an n ≥ 3 such that Σn-Subset implies ∆n+1-Subset over RCA0; the general status of this implication is unknown.

Background

The paper surveys known implications and separations among Γ-Subset principles. It is known that ∆2-Subset does not imply Σ2-Subset (Theorem 7.1), and results imply ∆n-Subset does not imply ∆n+1-Subset for all n ≥ 2.

They remark that certain non-implications do not straightforwardly relativize, leaving gaps for higher n. In particular, whether Σn-Subset might imply ∆n+1-Subset for some n ≥ 3 remains unsettled.