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The Jordan curve theorem and the Schönflies theorem in weak second-order arithmetic

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Abstract

In this paper, we show within \({\mathsf{RCA}_0}\) that both the Jordan curve theorem and the Schönflies theorem are equivalent to weak König’s lemma. Within \({\mathsf {WKL}_0}\) , we prove the Jordan curve theorem using an argument of non-standard analysis based on the fact that every countable non-standard model of \({\mathsf {WKL}_0}\) has a proper initial part that is isomorphic to itself (Tanaka in Math Logic Q 43:396–400, 1997).

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Authors and Affiliations

  1. Advanced Media, Inc., 48F Sunshine 60 Building, 3-1-1 Higashi-Ikebukuro, Toshima-ku, Tokyo, 170-6048, Japan

    Nobuyuki Sakamoto

  2. Mathematical Institute, Tohoku University, Sendai, 980-8578, Japan

    Keita Yokoyama

Authors
  1. Nobuyuki Sakamoto
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  2. Keita Yokoyama
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Correspondence to Nobuyuki Sakamoto.

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Sakamoto, N., Yokoyama, K. The Jordan curve theorem and the Schönflies theorem in weak second-order arithmetic. Arch. Math. Logic 46, 465–480 (2007). https://doi.org/10.1007/s00153-007-0050-6

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  • DOI: https://doi.org/10.1007/s00153-007-0050-6

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