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Memoirs of a proof theorist. Gödel and other logicians. Transl. from the 1998 revised Japanese edition by Mariko Yasugi and Nicholas Passell. (English) Zbl 1030.01030

River Edge, NJ: World Scientific. xviii, 135 p. (2003).
The book under review has to be turned to its back cover for realizing that its author is G. Takeuti. The persons mentioned on the front cover have translated and edited this collection of essays. It was originally published in Japanese under the title “Gödel” in 1986. The translation is from the revised and enlarged second edition of 1998. Some of the portions of the Japanese original have been omitted on the author’s request, some chapters have been added for the translation, such as Appendix B on Takeuti’s main result, the birth of second order proof theory by his Fundamental Conjecture on the Generalized Logic Calculus (GLC).
The essays give Takeuti’s very personal view on foundational studies in the post World War II period, and their pre-history. They furthermore reflect on his – scientific and personal – relations to eminent logicians of this period, above all K. Gödel, but also P. Bernays, A. A. Fraenkel, P. Erdős, A. Tarski, A. Heyting, A. Church, S. C. Kleene, and G. Kreisel.
Ch. 1 relates Takeuti’s fundamental conjecture to the work of K. Gödel. Ch. 2 not only discusses Gödel’s contributions to logic and foundations, but also hints at the significance of the work of P. Bernays. Ch. 3 discusses the relation between Gödel and Hilbert, arguing for the thesis that “Gödel’s academic career was molded by the goal of exceeding Hilbert” (p. 29). Ch. 4 gives short biographies with personal recollections of the logicians mentioned above. Ch. 5 presents some visions on the future of set theory, discussing, besides other topics, the Axiom of Determinateness and the relation between set theory and computer science. Ch. 6 surveys Hilbert’s programme and Gödel’s role within it. Ch. 7 considers the fate of Hilbert’s second problem, i. e. , the axiomatization of arithmetic and the proof of its consistency. Ch. 8 reports critically on the conference “Gödel ’96” in Brno, on the occasion of Gödel’s 90th birthday [see the proceedings, ed. P. Hájek, Berlin: Springer (1996; Zbl 0844.00017, reprinted 2001; Zbl 0961.03005)]. Ch. 9 reviews favorably the volume “Gödel remembered” [eds. P. Weingartner and L. Schmetterer, Napoli: Bibliopolis (1983; Zbl 0621.01002)]. Ch. 10 reprints “A Tribute to the Memory of Professor Gödel”. The book is closed by two appendices with technical papers, the first on Gödel’s Continuum Hypothesis, according to which the power of the continuum is \(\aleph_2\), the second on Takeuti’s Fundamental Conjecture already mentioned.

MSC:

01A70 Biographies, obituaries, personalia, bibliographies
01A60 History of mathematics in the 20th century
01-02 Research exposition (monographs, survey articles) pertaining to history and biography
03A05 Philosophical and critical aspects of logic and foundations
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