[logic-ml] Kobe Colloquium (talk by Sam Sanders)

[Makoto Kikuchi]

Kobe Colloquium on Logic, Statistics and Informatics

以下の要領でコロクウィウムを開催します。

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.

交通：阪急六甲駅またはJR六甲道駅から神戸市バス36系統「鶴甲団地」
行きに乗車，「神大本部工学部前」停留所下車，徒歩すぐ．
http://www.kobe-u.ac.jp/info/access/rokko/rokkodai-dai2.htm

連絡先：菊池誠　mkikuchi at kobe-u.ac.jp