A characterization of ordinal analysis

Abstract

Ordinal analysis induces a partition of ${\Sigma }_1^1$-definable and ${\Pi }_1^1$-sound theories whereby two theories are equivalent if they have the same proof-theoretic ordinal. We show that no equivalence relation $\equiv$ is finer than the ordinal analysis partition if both: (1) $T\equiv U$ whenever $T$ and $U$ prove the same ${\Pi }_1^1$ sentences; (2) $T\equiv T+U$ for every set $U$ of true ${\Sigma }_1^1$ sentences. In fact, no such equivalence relation makes a single distinction that the ordinal analysis partition does not make.

arXiv commentThis article has been merged with 2201.05284 to form 2209.09765