, Volume 46, Issue 5-6, pp 465-480
Date: 21 Mar 2007

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).