Characterizations of ordinal analysis

Ordinal analysis is a research program wherein recursive ordinals are assigned to axiomatic theories. According conventional wisdom, ordinal measures the strength of Yet what attendant notion strength? In this paper we present abstract characterizations that address question. First, characterize as partition Σ11-definable and Π11-sound theories, namely, whereby two theories equivalent if they have same proof-theoretic ordinal. We show no equivalence relation ≡ finer than both: (1) T≡U whenever T U prove Π11 sentences; (2) T≡T+U for every set true Σ11 sentences. fact, such makes single distinction does not make. Second, an ordering on arithmetically-definable T⩽U less or equal U. The standard ways measuring consistency inclusion Π10 theorems. introduce analogues these notions—Π11-reflection theorems—in presence oracle truths, coincide with induced by analysis.