Set Theory, Arithmetic, and Foundations of Mathematics

Theorems, Philosophies

Juliette Kennedy, Roman Kossak

• Publisher: Cambridge University Press
• Year: 2011
• Print ISBN: 9781107008045
• eBook ISBN: 9781139142816
• Topic: Mathematics
• Subtopic: Logic in Mathematics
• Index: 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 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arithmetics as well as on other aspects of arithmetics, such as interpretability; the transcript of Gödel's previously unpublished 1972–1975 conversations with Sue Toledo, along with an appreciation of the same by Curtis Franks; Hugh Woodin's paper arguing against the generic multiverse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.

Table of contents

• Cover
• Title
• Copyright
• Dedication
• Contents
• Introduction
• Historical remarks on Suslin’s problem
• The continuum hypothesis, the generic-multiverse of sets, and the Ω conjecture
• ω-models of finite set theory
• Tennenbaum’s theorem for models of arithmetic
• Hierarchies of subsystems of weak arithmetic
• Diophantine correct open induction
• Tennenbaum’s theorem and recursive reducts
• History of constructivism in the 20th century
• A very short history of ultrafinitism
• Sue Toledo’s notes of her conversations with Gödel in 1972–5
• Stanley Tennenbaum’s Socrates
• Tennenbaum’s proof of the irrationality of v2