Set theory, arithmetic, and foundations of mathematics
Subtitle: theorems, philosophies
By: edited by Juliette Kennedy, Roman Kossak
Series: Lecture notes in logic -- 36
Publisher: Cambridge University Press
Published: 2012
Place of publication: Cambridge; New York
Language: eng
ISBN-13: 9781107008045
Subjects: MATHEMATICS / Logic; Set theory; Philosophy; Symbolic and mathematical Logic; Mathematics
Description: "This collection of papers from various areas of mathematical logic showcases the remarkable breadth and richness of the field. Leading authors reveal how contemporary technical results touch upon foundational questions about the nature of mathematics. Highlights of the volume include: a history of Tennenbaum's theorem in arithmetic; a number of papers on Tennenbaum phenomena in weak arithmetics as well as on other aspects of arithmetics, such as interpretability; the transcript of Gödel's previously unpublished 1972-1975 conversations with Sue Toledo, along with an appreciation of the same by Curtis Franks; Hugh Woodin's paper arguing against the generic multiverse view; Anne Troelstra's history of intuitionism through 1991; and Aki Kanamori's history of the Suslin problem in set theory. The book provides a historical and philosophical treatment of particular theorems in arithmetic and set theory, and is ideal for researchers and graduate students in mathematical logic and philosophy of mathematics"--

"The papers collected here engage each of these questions through the veil of particular technical results. For example, the new proof of the irrationality of the square root of two, given by Stanley Tennenbaum in the 1960s and included here, brings into relief questions about the role simplicity plays in our grasp of mathematical proofs. In 1900 Hilbert asked a question which was not given at the Paris conference but which has been recently found in his notes for the list: find a criterion of simplicity in mathematics. The Tennenbaum proof is a particularly striking example of the phenomenon Hilbert contemplated in his 24th Problem"--
Table of contents: title: Machine generated contents note: 1. Introduction Juliette Kennedy and Roman Kossak; 2. Historical remarks on Suslin's problem Akihiro Kanamori; 3. The continuum hypothesis, the generic-multiverse of sets, and the [OMEGA] conjecture W. Hugh Woodin; 4. [omega]-Models of finite set theory Ali Enayat, James H. Schmerl and Albert Visser; 5. Tennenbaum's theorem for models of arithmetic Richard Kaye; 6. Hierarchies of subsystems of weak arithmetic Shahram Mohsenipour; 7. Diophantine correct open induction Sidney Raffer; 8. Tennenbaum's theorem and recursive reducts James H. Schmerl; 9. History of constructivism in the 20th century A. S. Troelstra; 10. A very short history of ultrafinitism Rose M. Cherubin and Mirco A. Mannucci; 11. Sue Toledo's notes of her conversations with Gödel in 1972-1975 Sue Toledo; 12. Stanley Tennenbaum's Socrates Curtis Franks; 13. Tennenbaum's proof of the irrationality of [the square root of] 2́.