The Jordan curve theorem and the Schönflies theorem in weak second-order arithmetic
Archive for Mathematical Logic 46 (5-6):465-480 (2007)
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)DOI
10.1007/s00153-007-0050-6
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Citations of this work
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References found in this work
Which set existence axioms are needed to prove the cauchy/peano theorem for ordinary differential equations?Stephen G. Simpson - 1984 - Journal of Symbolic Logic 49 (3):783-802.
The self-embedding theorem of WKL0 and a non-standard method.Kazuyuki Tanaka - 1997 - Annals of Pure and Applied Logic 84 (1):41-49.
On Formalization of Model-Theoretic Proofs of Gödel's Theorems.Makoto Kikuchi & Kazuyuki Tanaka - 1994 - Notre Dame Journal of Formal Logic 35 (3):403-412.
