The Foundations of Mathematics in the Theory

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
Preliminaries:
The idea of foundations of mathematics
Simple arithmetic
Basic Set Theory:
Semantics, ontology and logic
The principal axioms and definitions of set theory
Cantorian Set Theory:
Cantorian finitism
The axiomatic method
Axiomatic set theory
Euclidean Set Theory:
Euclidian finitism
The Euclidean theory of cardinality
The theory of simply infinite systems
Euclidean set theory from the Cantorian standpoint
Envoi
Appendices
Bibliography
Index
Table of Contents provided by Publisher. All Rights Reserved.

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The Foundations of Mathematics in the Theory of Sets

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Table of Contents

Supplemental Materials