Derivability conditions and Löb’s theorem for OGA

Determine whether the internal distribution and internal -completeness derivability conditions hold for the OGA provability predicate, and consequently whether Löb’s theorem is available in Omega Grounded Arithmetic (OGA).

Background

The paper proves that OGA derivability is recursively enumerable and that its provability predicate satisfies a meta-level characterization relating provability of a sentence to provability of its encoded provability claim. However, the authors explicitly distinguish this result from the remaining Hilbert–Bernays-style derivability conditions: internal distribution of provability over implication and internal Sigma-1 completeness are not established.

The unresolved issue is structurally important because, if the relevant derivability conditions hold, Löb’s theorem could become available in OGA. The paper notes that combining such a theorem with an internal soundness schema would threaten consistency, so determining exactly which derivability principles OGA supports is central to understanding its self-verification behavior.