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.

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单调和共最终超限归纳的序数分析

我们考虑了超限归纳法的两种变体，一种在谓词上具有单调性假设，另一种仅在下面共同定义时才具有归纳假设。后者可以看作是后继归纳的超限类似物，而通常的超限归纳是累积归纳。我们计算\（\ varDelta _0 \）公式的这些模式可在\（\ mathbf {I} \ varvec {\ Sigma} _n \）中证明的普通形式的最上值。通过基数2的幂，它大于证明理论序数\（| \ mathbf {I} \ varvec {\ Sigma} _n | \）。我们还显示了结构超限归纳的相似结果，定义为基本序列。