Overview
- Provides a comprehensive introduction to the sophisticated technique of forcing
- Includes Shelah’s astonishing construction of a model in which exactly 27 Ramsey ultrafilters exist
- Offers topics and open problems for further study
- Includes supplementary material: sn.pub/extras
Part of the book series: Springer Monographs in Mathematics (SMM)
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About this book
This book, now in a thoroughly revised second edition, provides a comprehensive and accessible introduction to modern set theory.
Following an overview of basic notions in combinatorics and first-order logic, the author outlines the main topics of classical set theory in the second part, including Ramsey theory and the axiom of choice. The revised edition contains new permutation models and recent results in set theory without the axiom of choice. The third part explains the sophisticated technique of forcing in great detail, now including a separate chapter on Suslin’s problem. The technique is used to show that certain statements are neither provable nor disprovable from the axioms of set theory. In the final part, some topics of classical set theory are revisited and further developed in light of forcing, with new chapters on Sacks Forcing and Shelah’s astonishing construction of a model with finitely many Ramsey ultrafilters.
Written for graduate students in axiomatic set theory, Combinatorial Set Theory will appeal to all researchers interested in the foundations of mathematics. With extensive reference lists and historical remarks at the end of each chapter, this book is suitable for self-study.
Following an overview of basic notions in combinatorics and first-order logic, the author outlines the main topics of classical set theory in the second part, including Ramsey theory and the axiom of choice. The revised edition contains new permutation models and recent results in set theory without the axiom of choice. The third part explains the sophisticated technique of forcing in great detail, now including a separate chapter on Suslin’s problem. The technique is used to show that certain statements are neither provable nor disprovable from the axioms of set theory. In the final part, some topics of classical set theory are revisited and further developed in light of forcing, with new chapters on Sacks Forcing and Shelah’s astonishing construction of a model with finitely many Ramsey ultrafilters.
Written for graduate students in axiomatic set theory, Combinatorial Set Theory will appeal to all researchers interested in the foundations of mathematics. With extensive reference lists and historical remarks at the end of each chapter, this book is suitable for self-study.
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Keywords
- axiom of choice
- Banach-Tarski paradox
- cardinal characteristics
- combinatorics of forcing
- forcing constructions
- forcing technique
- infinite combinatorics
- Martin's axiom
- permutation models
- Ramsey theory
- Ramsey ultrafilters
- set theory
- Suslin's problem
- MSC (2010): 03E35, 03E17, 03E25, 05D10, 03E30, 03E50, 03E05
- combinatorics
Table of contents (29 chapters)
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Front Matter
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Preliminary
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Front Matter
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Topics in Combinatorial Set Theory
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Front Matter
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Part III
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Front Matter
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Reviews
“Each chapter ends with Notes that often add historical information, offer further remarks on the chapter’s contents … . Halbheisen’s Combinatorial Set Theory is an excellent source for the intermediate or advanced student of set theory … Because of its wealth of material, it should also serve as an excellent resource for those designing advanced courses or searching for seminar assignments for students.” (J. M. Plotkin, zbMATH 06755662, 2018)
Authors and Affiliations
Bibliographic Information
Book Title: Combinatorial Set Theory
Book Subtitle: With a Gentle Introduction to Forcing
Authors: Lorenz J. Halbeisen
Series Title: Springer Monographs in Mathematics
DOI: https://doi.org/10.1007/978-3-319-60231-8
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer International Publishing AG 2017
Hardcover ISBN: 978-3-319-60230-1Published: 11 January 2018
Softcover ISBN: 978-3-319-86812-7Published: 04 June 2019
eBook ISBN: 978-3-319-60231-8Published: 20 December 2017
Series ISSN: 1439-7382
Series E-ISSN: 2196-9922
Edition Number: 2
Number of Pages: XVI, 594
Number of Illustrations: 20 b/w illustrations
Additional Information: Previous edition published by Springer-Verlag London, 2012