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Intuitionistic Provability versus Uniform Provability in \(\mathsf{RCA}\)

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Evolving Computability (CiE 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9136))

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Abstract

We provide an exact formalization of uniform provability in \(\mathsf {RCA}\) and show that for any \(\Pi _{2}^{1}\) sentence of some syntactical form, it is intuitionistically provable if and only if it is uniformly provable in \(\mathsf {RCA}\).

M. Fujiwara—The author is supported by a Grant-in-Aid for JSPS fellows.

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Notes

  1. 1.

    Troelstra considers \(\mathsf {HA}+ \mathrm{ECT_{0}}\)(extended Church’s thesis)\(+\mathrm{MP_{PR}}\)(the fragment of \(\mathrm {MP}\) only for primitive recursive \(\alpha \)) to be a formalization of Markov-style constructive mathematics [18, 4.4.12].

  2. 2.

    The exhaustive comparison between these two is in [6].

  3. 3.

    Troelstra [17] indicates some analogy between Weihrauch’s computable analysis and constructive mathematics.

  4. 4.

    The detailed proof is in author’s phD thesis [7].

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Acknowledgment

The author is grateful to his supervisor Takeshi Yamazaki and also to Ulrich Kohlenbach for helpful discussion.

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Authors and Affiliations

  1. Mathematical Institute, Tohoku University, Tohoku, Japan

    Makoto Fujiwara

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  1. Makoto Fujiwara
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Correspondence to Makoto Fujiwara .

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Editors and Affiliations

  1. Department of Computer Science, Swansea University, Swansea, United Kingdom

    Arnold Beckmann

  2. University of Bucharest, Bucharest, Romania

    Victor Mitrana

  3. Sofia University, Sofia, Bulgaria

    Mariya Soskova

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Fujiwara, M. (2015). Intuitionistic Provability versus Uniform Provability in \(\mathsf{RCA}\) . In: Beckmann, A., Mitrana, V., Soskova, M. (eds) Evolving Computability. CiE 2015. Lecture Notes in Computer Science(), vol 9136. Springer, Cham. https://doi.org/10.1007/978-3-319-20028-6_19

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  • DOI: https://doi.org/10.1007/978-3-319-20028-6_19

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  • Online ISBN: 978-3-319-20028-6

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