This 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.
' ... this book is thought-provoking ... distinctive approach to the twin issues of mathematical ontology and mathematical foundations'. Australian Journal of Philosophy
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.
Series: Encyclopedia of Mathematics and Its Applications
Number Of Pages: 446
Published: 23rd April 2001
Publisher: Cambridge University Press
Country of Publication: GB
Dimensions (cm): 23.4 x 15.6
Weight (kg): 0.8