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

Article

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).

Keywords

Second order arithmetic Reverse mathematics The Jordan curve theorem The Schönflies theorem Non-standard analysis 

Mathematics Subject Classification (2000)

Primary 03B30 Secondary 03F35 26E35 

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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Advanced Media, Inc.Toshima-kuJapan
  2. 2.Mathematical InstituteTohoku UniversitySendaiJapan

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