Switch to: References

Add citations

You must login to add citations.
  1. Tanaka’s theorem revisited.Saeideh Bahrami - 2020 - Archive for Mathematical Logic 59 (7-8):865-877.
    Tanaka proved a powerful generalization of Friedman’s self-embedding theorem that states that given a countable nonstandard model \\) of the subsystem \ of second order arithmetic, and any element m of \, there is a self-embedding j of \\) onto a proper initial segment of itself such that j fixes every predecessor of m. Here we extend Tanaka’s work by establishing the following results for a countable nonstandard model \\ \)of \ and a proper cut \ of \:Theorem A. The (...)
    No categories
    Direct download (2 more)  
    Translate
     
     
    Export citation  
     
    Bookmark  
  • The Computational Content of Nonstandard Analysis.Sam Sanders - unknown
    Kohlenbach's proof mining program deals with the extraction of effective information from typically ineffective proofs. Proof mining has its roots in Kreisel's pioneering work on the so-called unwinding of proofs. The proof mining of classical mathematics is rather restricted in scope due to the existence of sentences without computational content which are provable from the law of excluded middle and which involve only two quantifier alternations. By contrast, we show that the proof mining of classical Nonstandard Analysis has a very (...)
    No categories
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark  
  • Nonstandard Second-Order Arithmetic and Riemannʼs Mapping Theorem.Yoshihiro Horihata & Keita Yokoyama - 2014 - Annals of Pure and Applied Logic 165 (2):520-551.
    In this paper, we introduce systems of nonstandard second-order arithmetic which are conservative extensions of systems of second-order arithmetic. Within these systems, we do reverse mathematics for nonstandard analysis, and we can import techniques of nonstandard analysis into analysis in weak systems of second-order arithmetic. Then, we apply nonstandard techniques to a version of Riemannʼs mapping theorem, and show several different versions of Riemannʼs mapping theorem.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  • The Jordan Curve Theorem and the Schönflies Theorem in Weak Second-Order Arithmetic.Nobuyuki Sakamoto & Keita Yokoyama - 2007 - Archive for Mathematical Logic 46 (5-6):465-480.
    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).
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   4 citations