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Nets and Reverse Mathematics

Some Initial Results

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Computing with Foresight and Industry (CiE 2019)

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Abstract

Nets are generalisations of sequences involving possibly uncountable index sets; this notion was introduced about a century ago by Moore and Smith, together with the generalisation to nets of various basic theorems of analysis due to Bolzano-Weierstrass, Dini, Arzelà, and others. This paper deals with the Reverse Mathematics study of theorems about nets indexed by subsets of Baire space, i.e. part of third-order arithmetic. Perhaps surprisingly, over Kohlenbach’s base theory of higher-order Reverse Mathematics, the Bolzano-Weierstrass theorem for nets and the unit interval implies the Heine-Borel theorem for uncountable covers. Hence, the former theorem is extremely hard to prove (in terms of the usual hierarchy of comprehension axioms), but also unifies the concepts of sequential and open-cover compactness. Similarly, Dini’s theorem for nets is equivalent to the uncountable Heine-Borel theorem.

This research was supported by the John Templeton Foundation grant a new dawn of intuitionism with ID 60842. Opinions expressed in this paper do not necessarily reflect those of the John Templeton Foundation. I thank Thomas Streicher and Anil Nerode for their valuable advice. Finally, I thank the three referees for their valuable suggestions that have significantly improved the paper.

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Notes

  1. 1.

    To be absolutely clear, variables (of any finite type) are allowed in quantifier-free formulas of the language : only quantifiers are banned.

  2. 2.

    Simpson mentions in [42] the caveat that e.g. and have the same first-order strength, but the latter is strictly stronger than the former.

  3. 3.

    There are some examples of theorems (predating and [35]) that fall outside of the Gödel hierarchy based on inclusion, like special cases of Ramsey’s theorem and the axiom of determinacy from set theory ([15, 23]). These are far less natural than e.g. Heine-Borel compactness, in our opinion.

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  1. School of Mathematics, Leeds University, Leeds, UK

    Sam Sanders

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  1. Sam Sanders
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  1. Christian Albrechts University of Kiel, Kiel, Germany

    Florin Manea

  2. Durham University, Durham, UK

    Barnaby Martin

  3. Durham University, Durham, UK

    Daniël Paulusma

  4. Università degli Studi di Milano, Milan, Italy

    Giuseppe Primiero

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Sanders, S. (2019). Nets and Reverse Mathematics. In: Manea, F., Martin, B., Paulusma, D., Primiero, G. (eds) Computing with Foresight and Industry. CiE 2019. Lecture Notes in Computer Science(), vol 11558. Springer, Cham. https://doi.org/10.1007/978-3-030-22996-2_22

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  • DOI: https://doi.org/10.1007/978-3-030-22996-2_22

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