The foundations of mathematics in the theory of sets
著者
書誌事項
The foundations of mathematics in the theory of sets
（Encyclopedia of mathematics and its applications / edited by G.-C. Rota, v. 82）
Cambridge University Press, 2000
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注記
Bibliography: p. 415-420
Includes index
内容説明・目次
内容説明
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.
目次
- 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.
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