The Foundations of Mathematics in the Theory of Sets (Encyclopedia of Mathematics and its Applications, Series Number 82)

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. • Written by a leading researcher in the field • Of interest to philosophers as well as mathematicians • At the time of publication, there were no other books that deal with the foundations of mathematics in such detail 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. Preliminaries -- Idea Of Foundations For Mathematics -- Simple Arithmetic -- Basic Set Theory -- Semantics, Ontology, And Logic -- Principal Axioms And Definitions Of Set Theory -- Cantorian Set Theory -- Cantorian Finitism -- Axiomatic Method -- Axiomatic Set Theory -- Euclidean Set Theory -- Euclidean Finitism -- Euclidean Theory Of Cardinality -- Euclidean Theory Of Simply Infinite Systems -- Euclidean Set Theory From The Cantorian Standpoint -- Envoi. J.p. Mayberry. Includes Bibliographical References (p. 415-420) And Index. This unified approach to the foundations of mathematics in the theory of sets covers 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". The book contains an investigation of the logic of quantification over the universe of sets and a discussion of its role in second order logic, and the analysis of proof by induction and definition by recursion. The book should appeal to both philosophers and mathematicians with an interest in the foundations of mathematics. This 2001 book will appeal to mathematicians and philosophers interested in the foundations of mathematics Mathematics differs from all the other sciences in requiring that its propositions be proved.