Kobe Colloquium on Logic, Statistics and Informatics

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日時：2011年1月20日（木）15:30 〜 場所：神戸大学自然科学総合研究棟3号館4階421室（渕野グループプレゼンテーション室） 講演者：Sam Sanders（東北大学） 題目：A copy of several Reverse Mathematics

アブストラクト：

Reverse Mathematics (RM) is a program in the foundations of mathematics initiated by Friedman ([1, 2]) and developed extensively by Simpson ([4]). The aim of RM is to determine which minimal axioms prove theorems of ordinary mathematics. The main theme of RM is that a theorem of ordinary mathematics is either provable in RCA_0, or the theorem is equivalent to either WKL_0, ACA_0, ATR_0 or Π^1_1-CA_0. This equivalence is proved in RCA_0, the ‘base theory’ of RM. Moreover, each of these systems corresponds to a well-known foundational program in mathematics.

Nonstandard Analysis has played an important role in RM ([3, 5]). We are interested in RM where equality is replaced by the relation ≈, i.e. equality up to infinitesimals. We obtain a ‘copy’ of the RM of WKL_0 and ACA_0 in a weak system of Nonstandard Analysis. Surprisingly, the same system is also a ‘copy’ of Constructive Reverse Mathematics. Our results have applications in Physics, Theoretical Computer Science, and the Philosophy of Science.

References [1] Harvey Friedman, Some systems of second order arithmetic and their use, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), vol. 1, Canad. Math. Congress, 1975, pp. 235–242. [2] Harvey Friedman, Systems of second order arithmetic with restricted induction I & II, Journal of Symbolic Logic, vol. 41 (1976), pp. 557–559. [3] H. Jerome Keisler, Nonstandard arithmetic and reverse mathematics, Bulletin of Symbolic Logic, vol. 12 (2006), no. 1, pp. 100–125. [4] Stephen G. Simpson, Subsystems of second order arithmetic, Perspectives in Mathematical Logic, Springer–Verlag, Berlin, 1999. [5] Kazuyuki Tanaka, The self-embedding theorem of WKL_0 and a non- standard method, Annals of Pure and Applied Logic, vol. 84 (1997), pp. 41–49.

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連絡先：菊池誠　[email protected]