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Nonstandard second-order arithmetic and Riemann’s mapping theorem. (English) Zbl 1298.03112

The paper is devoted to the development of nonstandard analysis in systems of nonstandard second-order arithmetic. Some nonstandard axioms like the standard part principle and the transfer principle are used. In this way some nonstandard proofs in ns-ACA\(_{0}\) and ns-WKL\(_{0}\) for some standard theorems are done. Also reverse mathematics for nonstandard analysis is done – more exactly, reverse mathematics for some nonstandard counterparts of standard theorems. A nonstandard technique is applied to a version of Riemann’s mapping theorem. It is shown that Riemann’s mapping theorem for a polygonal domain is provable within RCA\(_{0}\) and that for a Jordan region it is equivalent to WKL\(_{0}\).

MSC:

03F35 Second- and higher-order arithmetic and fragments
03B30 Foundations of classical theories (including reverse mathematics)
03H15 Nonstandard models of arithmetic
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References:

[1] Ahlfors, L. V., Complex Analysis (1966), McGraw-Hill · Zbl 0154.31904
[2] Goldblatt, Robert, Lectures on the Hyperreals: An Introduction to Nonstandard Analysis, Grad. Texts in Math., vol. 188 (1998), Springer-Verlag, XIV+289 pp · Zbl 0911.03032
[3] Horihata, Yoshihiro, Subsystems of first and second order arithmetic (August 2011), Tohoku University, Doctoral thesis
[4] Keisler, H. Jerome, Nonstandard arithmetic and reverse mathematics, Bull. Symbolic Logic, 12, 1, 100-125 (2006) · Zbl 1101.03040
[5] Keisler, H. Jerome, Nonstandard arithmetic and recursive comprehension, Ann. Pure Appl. Logic, 161, 8, 1047-1062 (2010) · Zbl 1230.03092
[6] Sakamoto, Nobuyuki; Yokoyama, Keita, The Jordan curve theorem and the Schönflies theorem in weak second-order arithmetic, Arch. Math. Logic, 46, 465-480 (July 2007)
[7] Sanders, Sam, The Reverse Mathematics of Elementary Recursive Nonstandard Analysis: A Robust Contribution to the Foundations of Mathematics (2010), Ghent University, PhD thesis
[8] Sanders, Sam, ERNA and Friedmanʼs reverse mathematics, J. Symbolic Logic, 76, 2, 637-664 (2011) · Zbl 1231.03059
[9] Simpson, Stephen G., Persp. Logic (2009), Association for Symbolic Logic, Cambridge University Press, XVI+444 pp
[10] Simpson, Stephen G.; Yokoyama, Keita, A nonstandard counterpart of WWKL, Notre Dame J. Form. Log., 52 (2011) · Zbl 1250.03118
[11] Tanaka, Kazuyuki, Non-standard analysis in \(W K L_0\), Math. Log. Q., 43, 3, 396-400 (1997) · Zbl 0888.03037
[12] Tanaka, Kazuyuki, The self-embedding theorem of \(W K L_0\) and a non-standard method, Ann. Pure Appl. Logic, 84, 41-49 (1997) · Zbl 0871.03044
[13] Tanaka, K.; Yamazaki, T., A non-standard construction of Haar measure and weak Königʼs lemma, J. Symbolic Logic, 65, 1, 173-186 (2000) · Zbl 0949.03056
[14] Yokoyama, Keita, Complex analysis in subsystems of second order arithmetic, Arch. Math. Logic, 46, 15-35 (2007) · Zbl 1112.03053
[15] Yokoyama, Keita, Non-standard analysis in \(ACA_0\) and Riemann mapping theorem, Math. Log. Q., 53, 2, 132-146 (2007) · Zbl 1115.03085
[16] Yokoyama, Keita, Formalizing non-standard arguments in second order arithmetic, J. Symbolic Logic, 75, 4, 1199-1210 (2010) · Zbl 1214.03047
[18] Yu, X.; Simpson, S. G., Measure theory and weak Königʼs lemma, Arch. Math. Logic, 30, 171-180 (1990) · Zbl 0718.03043
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