The Foundations of Mathematics in the Theory of Sets

John P. Mayberry

doi:10.1017/CBO9781139087124

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Sergeyev, Yaroslav D. 2010. Cellular Automata. Vol. 6350, Issue. , p. 646.

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Sergeyev, Yaroslav D. and Garro, Alfredo 2013. Single-tape and multi-tape Turing machines through the lens of the Grossone methodology. The Journal of Supercomputing, Vol. 65, Issue. 2, p. 645.

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Sergeyev, Yaroslav D. 2014. Frontiers of Fundamental Physics and Physics Education Research. Vol. 145, Issue. , p. 353.

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Sergeyev, Yaroslav D. and Garro, Alfredo 2015. Automata, Universality, Computation. Vol. 12, Issue. , p. 139.

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John P. Mayberry

Publisher:: Cambridge University Press
Online publication date:: March 2012
Print publication year:: 2001
Online ISBN:: 9781139087124
DOI:: https://doi.org/10.1017/CBO9781139087124

Subjects:: Mathematics (general), Mathematics, Logic, Categories and Sets, Philosophy: General Interest, Philosophy
Series:: Encyclopedia of Mathematics and its Applications (82)

62.99 (GBP)

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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.

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

Source: Australasian Journal of Philosophy

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The Foundations of Mathematics in the Theory of Sets

This Book has been cited by the following publications. This list is generated based on data provided by Crossref.

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Contents

Frontmatter
pp i-vi

Contents
pp vii-ix

Preface
pp x-xx

Part One - Preliminaries
pp 1-2

1 - The Idea of Foundations for Mathematics
pp 3-16

2 - Simple Arithmetic
pp 17-64

Part Two - Basic Set Theory
pp 65-66

3 - Semantics, Ontology, and Logic
pp 67-110

4 - The Principal Axioms and Definitions of Set Theory
pp 111-150

Part Three - Cantorian Set Theory
pp 151-152

5 - Cantorian Finitism
pp 153-190

6 - The Axiomatic Method
pp 191-236

7 - Axiomatic Set Theory
pp 237-258

Part Four - Euclidean Set Theory
pp 259-260

8 - Euclidean Finitism
pp 261-299

9 - The Euclidean Theory of Cardinality
pp 300-324

10 - The Euclidean Theory of Simply Infinite Systems
pp 325-368

11 - Euclidean Set Theory from the Cantorian Standpoint
pp 369-380

12 - Envoi
pp 381-395

Appendix 1 - Conceptual Notation
pp 396-410

Appendix 2 - The Rank of a Set
pp 411-414

Bibliography
pp 415-420

Index
pp 421-424

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The Foundations of Mathematics in the Theory of Sets

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