Takeuti Gaisi

Takeuti Gaisi ( Japanese 竹 内 外史 , Takeuchi Gaishi ; born January 25, 1926 in Kizu , Ishikawa Prefecture , Japan ; † May 10, 2017 ) was a Japanese mathematical logician, known for his contributions to proof theory .

Takeuti received his PhD in mathematical logic from the University of Tokyo in 1956 . From 1950 he was first assistant professor, later professor at the Tokyo University of Education and from 1966 professor at the University of Illinois at Urbana-Champaign . In 1996 he retired.

Takeuti's goal in the 1950s was to obtain evidence of consistency for formal systems that comprise analysis. For this purpose he further developed the methods of Gerhard Gentzen (among other things he introduced ordinal diagrams ).

In 1967 Takeuti succeeded in proving the absence of contradictions in analysis (the part of analysis in which the comprehension is restricted to arbitrary formulas with at most one set quantifier ). According to Kurt Schütte, this was the first proof of the absence of contradictions in an essentially impredicative part of mathematics and analysis. ${\displaystyle \Pi _{1}^{1}}$

Takeuti's conjecture from 1953 says that the rule of intersection applies in the logic calculus of finite order (see Gentzenscher's law ). It was proved for the second order calculus by William W. Tait (1966) and Dag Prawitz (1967, for higher order 1969) and independently from Takahashi Motoo (1967, also higher order) and Jean-Yves Girard .

Two textbooks and standard works came from Takeuti, one on proof theory and one on axiomatic set theory.

In the early 1950s he also dealt with homotopy theory and knot theory (he held a seminar in 1952/53 that his doctoral student Kunio Murasagi attended).

1959/60, 1966 to 1968 and 1971 to 1972 he was at the Institute for Advanced Study with Kurt Gödel .

From 2003 to 2009 he was President of the Kurt Gödel Society. In 1998 he received the Czech Bolzano Medal and the Okawa Prize for Publications. In 1982 he received the Asahi Prize .

Fonts

• Proof Theory . North Holland 1975, Dover 2013
• with Wilson M. Zaring: Introduction to Axiomatic Set Theory . Springer 1971, 1973 (as Axiomatic Set Theory)
• Two applications of logic to mathematics (= Publications of the Mathematical Society of Japan 13). Princeton University Press, Princeton (New Jersey), 1978
• Memoirs of a proof theorist. Godel and other logicians . World Scientific River Edge (New Jersey), 1998, 2003
• On a generalized logic calculus . In: Japanese Journal of Mathematics, Volume 23, 1953, pp. 39-96, Errata Volume 24, 1954, pp. 149-156
• Consistency proofs of some subsystems of analysis . In: Annals of Mathematics, Volume 86, 1967, pp. 299-348

Individual evidence

1. ↑ Life data according to American Men and Women of Science , Thomson Gale 2004.
   Norbert Preining: Gaisi Takeuti, 1926-2017 . Norbert Preining's website, May 12, 2017, accessed May 14, 2017
2. ↑ A term is called impredicative if it is only defined over an entity to which it belongs. Such impredicative concepts are common in analysis, for example the concept of the maximum of the values ​​of a function. The endeavor of Henri Poincaré and Bertrand Russell at the beginning of the 20th century was to avoid such terms in the foundation of mathematics.
   Kurt Schütte, Helmut Schwichtenberg : Mathematical Logic . In: Gerd Fischer (ed.): A century of mathematics: Festschrift for the anniversary of the DMV (= documents on the history of mathematics, 6). Vieweg, Braunschweig / Wiesbaden, 1990, ISBN 978-3-528-06326-9 , p. 726.
3. ↑ Kurt Schütte: Newer results of the proof theory . ( Memento of the original from December 28, 2013 in the Internet Archive ) Info: The archive link was automatically inserted and not yet checked. Please check the original and archive link according to the instructions and then remove this notice. ICM 1966 (pdf, 1.3 MB). 
4. ↑ A publication on the theory of homotopy was already published by J. Math. Soc. Japan accepted when he learned that George W. Whitehead had gotten ahead of him. Jozef H. Przytycki: Notes to the early history of the Knot Theory in Japan, 2001, Arxiv
5. ↑ Membership Book IAS 1980

Japanese name: As is common in Japan, the family name comes before the first name in this article . Thus Takeuti is the family name, Gaisi the first name.