Ordinal analyses for monotone and cofinal transfinite inductions

Abstract

We consider two variants of transfinite induction, one with monotonicity assumption on the predicate and one with the induction hypothesis only for cofinally many below. The latter can be seen as a transfinite analogue of the successor induction, while the usual transfinite induction is that of cumulative induction. We calculate the supremum of ordinals along which these schemata for $\varDelta _0$ formulae are provable in $\mathbf {I}\varvec{\Sigma }_n$. It is shown to be larger than the proof-theoretic ordinal $|\mathbf {I}\varvec{\Sigma }_n|$ by power of base 2. We also show a similar result for the structural transfinite induction, defined with fundamental sequences.