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  1. Number Theory and Elementary Arithmetic.Jeremy Avigad - 2003 - Philosophia Mathematica 11 (3):257-284.
    is a fragment of first-order aritlimetic so weak that it cannot prove the totality of an iterated exponential fimction. Surprisingly, however, the theory is remarkably robust. I will discuss formal results that show that many theorems of number theory and combinatorics are derivable in elementary arithmetic, and try to place these results in a broader philosophical context.
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  • A Non-Standard Construction of Haar Measure and Weak König's Lemma.Kazuyuki Tanaka & Takeshi Yamazaki - 2000 - Journal of Symbolic Logic 65 (1):173-186.
    In this paper, we show within RCA 0 that weak Konig's lemma is necessary and sufficient to prove that any (separable) compact group has a Haar measure. Within WKL 0 , a Haar measure is constructed by a non-standard method based on a fact that every countable non-standard model of WKL 0 has a proper initial part isomorphic to itself [10].
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  • Formalizing Non-Standard Arguments in Second-Order Arithmetic.Keita Yokoyama - 2010 - Journal of Symbolic Logic 75 (4):1199-1210.
    In this paper, we introduce the systems ns-ACA₀ and ns-WKL₀ of non-standard second-order arithmetic in which we can formalize non-standard arguments in ACA₀ and WKL₀, respectively. Then, we give direct transformations from non-standard proofs in ns-ACA₀ or ns-WKL₀ into proofs in ACA₀ or WKL₀.
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  • The Computational Content of Nonstandard Analysis.Sanders Sam - 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 (...)
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  • 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.
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  • Non-Standard Analysis in ACA0 and Riemann Mapping Theorem.Keita Yokoyama - 2007 - Mathematical Logic Quarterly 53 (2):132-146.
    This research is motivated by the program of reverse mathematics and non-standard arguments in second-order arithmetic. Within a weak subsystem of second-order arithmetic ACA0, we investigate some aspects of non-standard analysis related to sequential compactness. Then, using arguments of non-standard analysis, we show the equivalence of the Riemann mapping theorem and ACA0 over WKL0. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim).
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  • 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).
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