Knowledge, Proof and Dynamics pp 25-33 | Cite as

# A Logical Characterization of the Continuous Bar Induction

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## Abstract

The continuous bar induction (\(\mathrm {c}\text {-}\mathrm {BI}\)) is an instance of the monotone bar induction (\(\mathrm {{M}}\mathrm {BI} \)) which is constructively equivalent to the statement that every pointwise continuous function from \(\mathbb {N}^{\mathbb {N}}\) to \(\mathbb {N}\) is induced by an inductively generated neighborhood function. In this paper, we give a simple logical characterization of \(\mathrm {c}\text {-}\mathrm {BI}\) that \(\mathrm {c}\text {-}\mathrm {BI}\) is equivalent to the restriction of \(\mathrm {{M}}\mathrm {BI} \) to \(\Pi ^{0}_{1}\) bars over intuitionistic elementary analysis. We also show that the difference between \(\mathrm {c}\text {-}\mathrm {BI}\) and the classical bar induction (without monotonicity on the bar) can be captured by the lesser limited principle of omniscience (LLPO) in the special case when the side-predicates of bar induction are restricted to bounded \(\Sigma ^{0}_{2}\) predicates, respectively.

## Keywords

Bar induction Intuitionistic mathematics Neighborhood function Constructive reverse mathematics LLPO## MSC2010:

03F55 03F35 03B30 03B20## Notes

### Acknowledgements

We thank Josef Berger for pointing out a crucial error in an earlier draft of this paper. A part of this work had been carried out at Mathematisches Institut, LMU, München in August 2017. We are grateful to Helmut Schwichtenberg and Chuangjie Xu for their invitation and hospitality for our visit. The visit of the first author was supported by the Core-to-Core Program (A. Advanced Research Networks) of Japan Society for the Promotion of Science. This work is also supported by Waseda University Grant for Special Research Projects 2018K-461.

## References

- Akama, Yohji, Stefano Berardi, Susumu Hayashi, and Ulrich Kohlenbach. 2004. An arithmetical hierarchy of the law of excluded middle and related principles. In
*Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LICS’04)*, 192–201.Google Scholar - Berger, Josef. 2006. The logical strength of the uniform continuity theorem. In
*Logical Approaches to Computational Barriers, CiE 2006*, ed. Arnold Beckmann, Ulrich Berger, Benedikt Löwe, and John V. Tucker, 35–39, Lecture Notes in Computer Science Berlin: Springer.Google Scholar - Fujiwara, Makoto. 2015. Intuitionistic and uniform provability in reverse mathematics. PhD thesis, Tohoku University.Google Scholar
- Fujiwara, Makoto. 2019. Bar induction and restricted classical logic. In
*Logic, Language, Information, and Computation, WoLLIC 2019*, ed. Rosalie Iemhoff, Michael Moortgat, and Ruy de Queiroz, 236–247. Berlin, Heidelberg: Springer.Google Scholar - Howard, William A., and Georg Kreisel. 1966. Transfinite induction and bar induction of types zero and one, and the role of continuity in intuitionistic analysis.
*Journal of Symbolic Logic*31 (3): 325–358.Google Scholar - Ishihara, Hajime. 2005. Constructive reverse mathematics: compactness properties.
*From sets and types to topology and analysis*, 245–267, Oxford Logic Guides Oxford: Oxford University Press.Google Scholar - Kawai, Tatsuji. 2019. Principles of bar induction and continuity on Baire space.
*Journal of Logic and Analysis*11 (FT3): 1–20.Google Scholar - Kreisel, Georg, and Anne S. Troelstra. 1970. Formal systems for some branches of intuitionistic analysis.
*Annals of Mathematical Logic*1 (3): 229–387.Google Scholar - Troelstra, Anne S., and Dirk van Dalen. 1988.
*Constructivism in Mathematics: An Introduction. Volume I and II*. North-Holland.Google Scholar