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Some Nonstandard Equivalences in Reverse Mathematics

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Sailing Routes in the World of Computation (CiE 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10936))

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Abstract

Reverse Mathematics (RM) is a program in the foundations of mathematics founded by Friedman and developed extensively by Simpson. The aim of RM is finding the minimal axioms needed to prove a theorem of ordinary (i.e. non-set theoretical) mathematics. In the majority of cases, one also obtains an equivalence between the theorem and its minimal axioms. This equivalence is established in a weak logical system called the base theory; four prominent axioms which boast lots of such equivalences are dubbed mathematically natural by Simpson. In this paper, we show that a number of axioms from Nonstandard Analysis are equivalent to theorems of ordinary mathematics not involving Nonstandard Analysis. These equivalences are proved in a base theory recently introduced by van den Berg and the author. Our results combined with Simpson’s criterion for naturalness suggest the controversial point that Nonstandard Analysis is actually mathematically natural.

S. Sanders—This research was supported by the Alexander von Humboldt Foundation and LMU Munich (via the Excellence Initiative).

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Notes

  1. 1.

    Heine-Borel compactness was studied before 1895 by Cousin ([4, p. 22]). The collected works of Pincherle contain a footnote by the editors ([17, p. 67]) stating that the associated Teorema (from 1882) corresponds to the Heine-Borel theorem.

  2. 2.

    The Heine-Borel theorem in RM is restricted to countable covers ([22, IV.1]).

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Authors and Affiliations

  1. Center for Advanced Studies and Munich Center for Mathematical Philosophy, LMU Munich, Munich, Germany

    Sam Sanders

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  1. Sam Sanders
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Correspondence to Sam Sanders .

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Editors and Affiliations

  1. Kiel University, Kiel, Germany

    Florin Manea

  2. Queens College, CUNY, Queens, New York, USA

    Russell G. Miller

  3. Kiel University, Kiel, Germany

    Dirk Nowotka

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Sanders, S. (2018). Some Nonstandard Equivalences in Reverse Mathematics. In: Manea, F., Miller, R., Nowotka, D. (eds) Sailing Routes in the World of Computation. CiE 2018. Lecture Notes in Computer Science(), vol 10936. Springer, Cham. https://doi.org/10.1007/978-3-319-94418-0_37

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  • DOI: https://doi.org/10.1007/978-3-319-94418-0_37

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  • Online ISBN: 978-3-319-94418-0

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