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.
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Notes
For bar induction in Intuitionism, this might destory bars: even if A(x) is a bar, \((\forall y\,{\unlhd }\,x)A(y)\) is not necessarily a bar.
This is one of the standard definitions of proof-theoretic ordinal. See the list of various standard definitions in [5, Section 1.2], among which the present definition is listed as \(|T|_{\mathrm{WO}}\).
The present article is related especially to [5, Remark 3.9].
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Acknowledgements
The author thanks Toshiyasu Arai for pointing out an error in the earlier version. He also thanks the anonymous referees for their invaluable suggestions. This publication was made possible through the support of a grant from the John Templeton Foundation (Grant No. 58229). The opinions expressed in this publication are those of the author and do not necessarily reflect the views of the John Templeton Foundation.
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Sato, K. Ordinal analyses for monotone and cofinal transfinite inductions. Arch. Math. Logic 59, 277–291 (2020). https://doi.org/10.1007/s00153-019-00688-5
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DOI: https://doi.org/10.1007/s00153-019-00688-5
Keywords
- Ordinal analysis
- Monotone transfinite induction
- Cofinal transfinite induction
- Structural transfinite induction
- Boundedness theorem