Archive for Mathematical Logic

, 46:465

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

Authors

    • Advanced Media, Inc.
  • Keita Yokoyama
    • Mathematical InstituteTohoku University
Article

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 arithmeticReverse mathematicsThe Jordan curve theoremThe Schönflies theoremNon-standard analysis

Mathematics Subject Classification (2000)

Primary 03B30Secondary 03F3526E35

Copyright information

© Springer-Verlag 2007