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

References 1.

Aleksandrov P.S. (1956). Combinatorial Topology. Graylock Press, New York

MATH 2.

Berg G., Julian W., Mines R. and Richman F. (1975). The constructive Jordan curve theorem.

Rocky Mt. J. Math. 5: 225–236

MATH MathSciNet CrossRef 3.

Bertoglio N. and Chuaqui R. (1994). An elementary geometric nonstandard proof of the Jordan curve theorem.

Geometriae Dedicata 51: 14–27

CrossRef MathSciNet 4.

Brown, D.K.: Functional Analysis in Weak Subsystems of Second Order Arithmetic, Ph.D. Thesis, The Pennsylvania State University (1987)

5.

Friedman H. (1976). Systems of second order arithmetic with restricted induction, I, II.

J. Symb. Logic 41: 557–559

CrossRef 6.

Kanovei V. and Reeken M. (1998). A nonstandard proof of the Jordan curve theorem.

Real Anal. Exch. 24: 161–170

MATH MathSciNet 7.

Kikuchi M. and Tanaka K. (1994). On formalization of model-theoretic proofs of Gödel’s theorems.

Notre Dame J. Formal Logic 35: 403–412

MATH CrossRef MathSciNet 8.

Moise E.E. (1977). Geometric Topology in Dimensions 2 and 3. Springer, Heidelberg

MATH 9.

Narens L. (1971). A nonstandard proof of the Jordan curve theorem.

Pac. J. Math. 36: 219–229

MATH MathSciNet 10.

Shioji N. and Tanaka K. (1984). Fixed point theory in weak second order arithmetic.

J. Symb. Logic 47: 167–188

MathSciNet 11.

Simpson S.G. (1984). Which set existence axioms are needed to prove the Cauchy-Peano theorem for ordinary diffential equations?.

J. Symb. Logic 49: 783–802

MATH CrossRef MathSciNet 12.

Simpson S.G. (1999). Subsystems of Second Order Arithmetic. Perspectives in Mathematical Logic. Springer, Heidelberg

MATH 13.

Tanaka K. (1997). Non-standard analysis in WKLo.

Math. Logic Q. 43: 396–400

MATH 14.

Tanaka K. (1997). The self-embedding theorem of WKLo and a non-standard method.

Ann. Pure Appl. Logic 84: 41–49

MATH CrossRef MathSciNet 15.

Tanaka K. and Yamazaki T. (2000). A non-standard construction of Haar measure and weak König’s lemma.

J. Symb. Logic 65: 173–186

MATH CrossRef MathSciNet 16.

Tverberg H. (1980). A proof of the Jordan curve theorem.

Bull. Lond. Math. Soc. 12: 34–38

MATH CrossRef MathSciNet 17.

Velben O. (1905). Theory of plane curves in nonmetrical analysis situs. Trans. Am. Math. Soc. 12: 34–38

Authors and Affiliations 1. Advanced Media, Inc. Toshima-ku Japan 2. Mathematical Institute Tohoku University Sendai Japan