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Characterising Brouwer’s continuity by bar recursion on moduli of continuity

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Abstract

We identify bar recursion on moduli of continuity as a fundamental notion of constructive mathematics. We show that continuous functions from the Baire space \({{\mathbb {N}}}^{{\mathbb {N}}}\) to the natural numbers \({\mathbb {N}}\) which have moduli of continuity with bar recursors are exactly those functions induced by Brouwer operations. The connection between Brouwer operations and bar induction allows us to formulate several continuity principles on the Baire space stated in terms of bar recursion on continuous moduli which naturally characterise some variants of bar induction. These principles state that a certain kind of continuous function from \({{\mathbb {N}}}^{{\mathbb {N}}}\) to \({\mathbb {N}}\) admits a modulus of continuity with a bar recursor. The results for the Baire space are recast in the setting of the Cantor space \({{\{ 0,1 \}}}^{{\mathbb {N}}}\) using the notion of fan recursor, a bar recursor for binary trees. This yields characterisations of uniformly continuous functions on the Cantor space and fan theorem in terms of fan recursors. The results for the Cantor space hold over the extensional version of intuitionistic arithmetic in all finite types (\({\mathsf {E}}\text {-}\mathsf {HA}^\omega \)), and those for the Baire space hold over \({\mathsf {E}}\text {-}\mathsf {HA}^\omega \) extended with the type for Brouwer operations. Our work places Spector’s bar recursion in a proper context of Brouwer’s intuitionistic mathematics and clarifies the connection between bar recursion and bar induction.

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Notes

  1. Logically, the continuous bar induction is the restriction of the monotone bar induction to \(\Pi ^{0}_{1}\) bars [9].

  2. Throughout this paper, continuity means pointwise continuity; thus a function \(Y :{{\mathbb {N}}}^{{\mathbb {N}}}\rightarrow {\mathbb {N}}\) is continuous if and only if \( \forall \alpha ^{{{\mathbb {N}}}^{{\mathbb {N}}}} \exists n^{{\mathbb {N}}} \forall \beta ^{{{\mathbb {N}}}^{{\mathbb {N}}}} \left( \overline{\alpha }n = \overline{\beta }n \rightarrow Y\alpha = Y\beta \right) . \)

  3. A dialogue tree over the natural numbers is a labelled inductive tree in which each node is labelled by a natural number; see Sect. 3. For the notion of eloquent function, see [7, Section 2.2].

  4. Note that the assumption in [15, Theorem 2.4] that S be a bar, i.e. \(\forall \alpha \exists n S(\overline{\alpha }n)\), is not needed. In fact, this property follows from Lemma 4.9.

  5. Note that this statement holds over \({\mathsf {HA}}^{\omega }\) as its proof depends only on Lemma 4.9.

  6. By continuity principle, we mean a principle which identifies a weaker notion of continuity with a stronger one. Note that the principles introduced in this section are valid in classical mathematics since they are equivalent to some variants of bar induction (cf. Proposition 5.1 and Corollary 5.3). This is in contrast to the well-known intuitionistic principles such as \(\mathrm {WC}\text {-}\mathrm {N}\) and \(\mathrm {C}\text {-}\mathrm {N}\) [18, Section 4.6], which contradict classical mathematics.

  7. In Berger [2, 3]\(, \mathrm{CFT}\), \(\mathrm{DFT}\), and \(\mathrm {UC_{c}}\) are called \(\mathrm{c\text {-}FT}\), \(\mathrm{FAN}\), and \(\mathrm {MUC}\), respectively. We use different terminologies to keep the notation uniform and to avoid the conflict with the existing literature [17, 2.6.4], where \(\mathrm {MUC}\) is used for another principle.

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Acknowledgements

We thank Chuangjie Xu for helpful comments on an earlier version of this paper. Part of this work was carried out in May 2019 at the Zukunftskolleg of the University of Konstanz, which was hosting the first author as a visiting fellow. The authors thank the institute for their support and hospitality. This work was supported by JSPS KAKENHI Grant Numbers JP18K13450, JP19J01239, and JP20K14354.

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  1. School of Science and Technology, Meiji University, 1-1-1 Higashi-Mita, Tama-ku, Kawasaki-shi, Kanagawa, 214-8571, Japan

    Makoto Fujiwara

  2. Japan Advanced Institute of Science and Technology, 1-1 Asahidai, Nomi, Ishikawa, 923-1292, Japan

    Tatsuji Kawai

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  1. Makoto Fujiwara
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Fujiwara, M., Kawai, T. Characterising Brouwer’s continuity by bar recursion on moduli of continuity. Arch. Math. Logic 60, 241–263 (2021). https://doi.org/10.1007/s00153-020-00740-9

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