The Foundations of Mathematics in the Theory of Sets

This 2001 book will appeal to mathematicians and philosophers interested in the foundations of mathematics.

John P. Mayberry (Author)

9780521172714, Cambridge University Press

Paperback, published 27 October 2011

446 pages
23.4 x 15.6 x 2.3 cm, 0.62 kg

Review of the hardback: ' … this book is thought-provoking … distinctive approach to the twin issues of mathematical ontology and mathematical foundations'. Australasian Journal of Philosophy

This 2001 book presents a unified approach to the foundations of mathematics in the theory of sets, covering both conventional and finitary (constructive) mathematics. It is based on a philosophical, historical and mathematical analysis of the relation between the concepts of 'natural number' and 'set'. This leads to an investigation of the logic of quantification over the universe of sets and a discussion of its role in second order logic, as well as in the analysis of proof by induction and definition by recursion. The subject matter of the book falls on the borderline between philosophy and mathematics, and should appeal to both philosophers and mathematicians with an interest in the foundations of mathematics.

Preface
Part I. Preliminaries: 1. The idea of foundations of mathematics
2. Simple arithmetic
Part II. Basic Set Theory: 3. Semantics, ontology and logic
4. The principal axioms and definitions of set theory
Part III. Cantorian Set Theory: 5. Cantorian finitism
6. The axiomatic method
7. Axiomatic set theory
Part IV. Euclidean Set Theory: 8. Euclidian finitism
9. The Euclidean theory of cardinality
10. The theory of simply infinite systems
11. Euclidean set theory from the Cantorian standpoint
12. Envoi
Appendices
Bibliography
Index.

Subject Areas: Set theory [PBCH], Mathematical logic [PBCD], Philosophy [HP]