The Jordan curve theorem and the Schönflies theorem in weak second-order arithmetic Authors Nobuyuki Sakamoto Keita Yokoyama Mathematical Institute Tohoku University Article

First Online: 21 March 2007 Received: 10 October 2006 DOI :
10.1007/s00153-007-0050-6

Cite this article as: Sakamoto, N. & Yokoyama, K. Arch. Math. Logic (2007) 46: 465. doi:10.1007/s00153-007-0050-6
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

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