Set Theory, Arithmetic, and Foundations of Mathematics

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 Tennenbaums 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arithmetics as well as on other aspects of arithmetics, such as interpretability; the transcript of Gödels previously unpublished 1972–1975 conversations with Sue Toledo, along with an appreciation of the same by Curtis Franks; Hugh Woodins paper arguing against the generic multiverse view; Anne Troelstras history of intuitionism through 1991; and Aki Kanamoris 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.

Spis treści:
1. Introduction Juliette Kennedy and Roman Kossak
2. Historical remarks on Suslins problem Akihiro Kanamori
3. The continuum hypothesis, the generic-multiverse of sets, and the Ω conjecture W. Hugh Woodin
4. ω-Models of finite set theory Ali Enayat, James H. Schmerl and Albert Visser
5. Tennenbaums theorem for models of arithmetic Richard Kaye
6. Hierarchies of subsystems of weak arithmetic Shahram Mohsenipour
7. Diophantine correct open induction Sidney Raffer
8. Tennenbaums 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 Toledos notes of her conversations with Gödel in 1972–1975 Sue Toledo
12. Stanley Tennenbaums Socrates Curtis Franks
13. Tennenbaums proof of the irrationality of √2.