# Set Theory and the Continuum Problem

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## Description

Set Theory and the Continuum Problem is a novel introduction to set theory, including axiomatic development, consistency, and independence results. It is self-contained and covers all the set theory that a mathematician should know. This book is intended for graduate students and researchers in mathematical logic.

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## Product details

- Hardback | 301 pages
- 150 x 230 x 25.4mm | 572g
- 14 Nov 1996
- Oxford University Press
- Clarendon Press
- Oxford, United Kingdom
- English
- 2 line figures, bibliography
- 0198523955
- 9780198523956

## Table of contents

Part I. General background. Some basics of class-set theory. The natural number. Superinduction, well ordering and choice. Ordinal numbers. Order isomorphism and transfinite recursion. Rank. Foundation, e-induction, and rank. Cardinals. Part II. Mostowski-Shepherdson Mappings. Reflection principles. Constructible sets. L is well founded first-order universe. Constructability is absolute over L. Constructability and the continuum hypothesis. Part III. Forcing, the very idea. The construction of S4 models and ZF. The axion of constructability is independent. Independence of the continuum hypothesis. Independence of the axiom of choice. Constructing classical models. Forcing background. References. Subject Index. Notation Index

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## Review quote

"Smullyan and Fitting. . .achieve miraculous clarity in a subject crowded with intimidating espositions; in particular their book meets the very high standard of exposition set by Smullyan's previous works." --Choice

"This text is a general introduction to NBG (von Neumann-Bernays-Godel class-set theory), and to Godel and Cohen proofs of the relative consistency and the independence of the generalized continuum hypothesis (GCH) and the axiom of choice (AC). . . .The authors write with admirable lucidity. There are some truly charming set pieces on countability and uncountability and on mathematical induction--I intend to appropriate them for my classes. . . .this is an excellent book for anyone interested in set theory and foundations."--Mathematical Reviews

"A well-written discussion of set theory, and readers will need a solid background in mathematics to fully appreciate its contents. The book is self-contained and intended for advanced undergraduates and graduate students in mathematics and computer science, especially those interested in set theory and its relationship to logic." --Computing Reviews

"Intended as a text for advanced undergraduates and graduate students. Essentially self-contained."--The Bulletin of Mathematics Books

"The book under review is a textbook for a beginning graduate course on set theory. The structure is fairly standard, with the book divided into three main sections; after an introductory section developing the basic facts about the universe of set theory, there is a section on constructibility and a section on forcing. The main goals of the book are to give proofs that the axiom of choice (AC) and the generalised continuum hypothesis (GCH) are consistent with and independent of the axioms of Zermelo-Fraenkel set theory (ZF). . . . The distinctive features of this book are the use of class set theory, the treatment of induction, and the use of modal logic in the treatment of forcing. The writing is lucid and accurate, and the main theorems are proved in an efficient way."--Journal of Symbolic Logic

" "Smullyan and Fitting. . .achieve miraculous clarity in a subject crowded with intimidating espositions; in particular their book meets the very high standard of exposition set by Smullyan's previous works." --Choice

"This text is a general introduction to NBG (von Neumann-Bernays-Godel class-set theory), and to Godel and Cohen proofs of the relative consistency and the independence of the generalized continuum hypothesis (GCH) and the axiom of choice (AC). . . .The authors write with admirable lucidity. There are some truly charming set pieces on countability and uncountability and on mathematical induction--I intend to appropriate them for my classes. . . .this is an excellent book for anyone interested in set theory and foundations."--Mathematical Reviews

"A well-written discussion of set theory, and readers will need a solid background in mathematics to fully appreciate its contents. The book is self-contained and intended for advanced undergraduates and graduate students in mathematics and computer science, especially those interested in set theory and its relationship to logic." --Computing Reviews

"Intended as a text for advanced undergraduates and graduate students. Essentially self-contained."--The Bulletin of Mathematics Books

"The book under review is a textbook for a beginning graduate course on set theory. The structure is fairly standard, with the book divided into three main sections; after an introductory section developing the basic facts about the universe of set theory, there is a section on constructibility and a section on forcing. The main goals of the book are to give proofs that the axiom of choice (AC) and the generalised continuumhypothesis (GCH) are consistent with and independent of the axioms of Zermelo-Fraenkel set theory (ZF). . . . The distinctive features of this book are the use of class set theory, the treatment of induction, and the use of modal logic in the treatment of forcing. The writing is lucid and accurate, and the main theorems are proved in an efficient way."--Journal of Symbolic Logic "Smullyan and Fitting. . .achieve miraculous clarity in a subject crowded with intimidating espositions; in particular their book meets the very high standard of exposition set by Smullyan's previous works." --Choice

"This text is a general introduction to NBG (von Neumann-Bernays-Godel class-set theory), and to Godel and Cohen proofs of the relative consistency and the independence of the generalized continuum hypothesis (GCH) and the axiom of choice (AC). . . .The authors write with admirable lucidity. There

are some truly charming set pieces on countability and uncountability and on mathematical induction--I intend to appropriate them for my classes. . . .this is an excellent book for anyone interested in set theory and foundations."--Mathematical Reviews

"A well-written discussion of set theory, and readers will need a solid background in mathematics to fully appreciate its contents. The book is self-contained and intended for advanced undergraduates and graduate students in mathematics and computer science, especially those interested in set theory

and its relationship to logic." --Computing Reviews

"Intended as a text for advanced undergraduates and graduate students. Essentially self-contained."--The Bulletin of Mathematics Books

"The book under review is a textbook for a beginning graduate course on set theory. The structure is fairly standard, with the book divided into three main sections; after an introductory section developing the basic facts about the universe of set theory, there is a section on constructibility and

a section on forcing. The main goals of the book are to give proofs that the axiom ofchoice (AC) and the generalised continuum hypothesis (GCH) are consistent with and independent of the axioms of Zermelo-Fraenkel set theory (ZF). . . . The distinctive features of this book are the use of class set

theory, the treatment of induction, and the use of modal logic in the treatment of forcing. The writing is lucid and accurate, and the main theorems are proved in an efficient way."--Journal of Symbolic Logic "Smullyan and Fitting. . .achieve miraculous clarity in a subject crowded with intimidating espositions; in particular their book meets the very high standard of exposition set by Smullyan's previous works." --Choice

"This text is a general introduction to NBG (von Neumann-Bernays-Godel class-set theory), and to Godel and Cohen proofs of the relative consistency and the independence of the generalized continuum hypothesis (GCH) and the axiom of choice (AC). . . .The authors write with admirable lucidity. There

are some truly charming set pieces on countability and uncountability and on mathematical induction--I intend to appropriate them for my classes. . . .this is an excellent book for anyone interested in set theory and foundations."--Mathematical Reviews

"A well-written discussion of set theory, and readers will need a solid background in mathematics to fully appreciate its contents. The book is self-contained and intended for advanced undergraduates and graduate students in mathematics and computer science, especially those interested in set theory

and its relationship to logic." --Computing Reviews

"Intended as a text for advanced undergraduates and graduate students. Essentially self-contained."--The Bulletin of Mathematics Books

"The book under review is a textbook for a beginning graduate course on set theory. The structure is fairly standard, with the book divided into three main sections; after an introductory section developing the basic facts about the universe of set theory, there is a section on constructibility and

a section on forcing. The main goals of the book are to give proofs that the axiom of choice (AC) and the generalised continuumhypothesis (GCH) are consistent with and independent of the axioms of Zermelo-Fraenkel set theory (ZF). . . . The distinctive features of this book are the use of class set

theory, the treatment of induction, and the use of modal logic in the treatment of forcing. The writing is lucid and accurate, and the main theorems are proved in an efficient way."--Journal of Symbolic Logic

show more

"This text is a general introduction to NBG (von Neumann-Bernays-Godel class-set theory), and to Godel and Cohen proofs of the relative consistency and the independence of the generalized continuum hypothesis (GCH) and the axiom of choice (AC). . . .The authors write with admirable lucidity. There are some truly charming set pieces on countability and uncountability and on mathematical induction--I intend to appropriate them for my classes. . . .this is an excellent book for anyone interested in set theory and foundations."--Mathematical Reviews

"A well-written discussion of set theory, and readers will need a solid background in mathematics to fully appreciate its contents. The book is self-contained and intended for advanced undergraduates and graduate students in mathematics and computer science, especially those interested in set theory and its relationship to logic." --Computing Reviews

"Intended as a text for advanced undergraduates and graduate students. Essentially self-contained."--The Bulletin of Mathematics Books

"The book under review is a textbook for a beginning graduate course on set theory. The structure is fairly standard, with the book divided into three main sections; after an introductory section developing the basic facts about the universe of set theory, there is a section on constructibility and a section on forcing. The main goals of the book are to give proofs that the axiom of choice (AC) and the generalised continuum hypothesis (GCH) are consistent with and independent of the axioms of Zermelo-Fraenkel set theory (ZF). . . . The distinctive features of this book are the use of class set theory, the treatment of induction, and the use of modal logic in the treatment of forcing. The writing is lucid and accurate, and the main theorems are proved in an efficient way."--Journal of Symbolic Logic

" "Smullyan and Fitting. . .achieve miraculous clarity in a subject crowded with intimidating espositions; in particular their book meets the very high standard of exposition set by Smullyan's previous works." --Choice

"This text is a general introduction to NBG (von Neumann-Bernays-Godel class-set theory), and to Godel and Cohen proofs of the relative consistency and the independence of the generalized continuum hypothesis (GCH) and the axiom of choice (AC). . . .The authors write with admirable lucidity. There are some truly charming set pieces on countability and uncountability and on mathematical induction--I intend to appropriate them for my classes. . . .this is an excellent book for anyone interested in set theory and foundations."--Mathematical Reviews

"A well-written discussion of set theory, and readers will need a solid background in mathematics to fully appreciate its contents. The book is self-contained and intended for advanced undergraduates and graduate students in mathematics and computer science, especially those interested in set theory and its relationship to logic." --Computing Reviews

"Intended as a text for advanced undergraduates and graduate students. Essentially self-contained."--The Bulletin of Mathematics Books

"The book under review is a textbook for a beginning graduate course on set theory. The structure is fairly standard, with the book divided into three main sections; after an introductory section developing the basic facts about the universe of set theory, there is a section on constructibility and a section on forcing. The main goals of the book are to give proofs that the axiom of choice (AC) and the generalised continuumhypothesis (GCH) are consistent with and independent of the axioms of Zermelo-Fraenkel set theory (ZF). . . . The distinctive features of this book are the use of class set theory, the treatment of induction, and the use of modal logic in the treatment of forcing. The writing is lucid and accurate, and the main theorems are proved in an efficient way."--Journal of Symbolic Logic "Smullyan and Fitting. . .achieve miraculous clarity in a subject crowded with intimidating espositions; in particular their book meets the very high standard of exposition set by Smullyan's previous works." --Choice

"This text is a general introduction to NBG (von Neumann-Bernays-Godel class-set theory), and to Godel and Cohen proofs of the relative consistency and the independence of the generalized continuum hypothesis (GCH) and the axiom of choice (AC). . . .The authors write with admirable lucidity. There

are some truly charming set pieces on countability and uncountability and on mathematical induction--I intend to appropriate them for my classes. . . .this is an excellent book for anyone interested in set theory and foundations."--Mathematical Reviews

"A well-written discussion of set theory, and readers will need a solid background in mathematics to fully appreciate its contents. The book is self-contained and intended for advanced undergraduates and graduate students in mathematics and computer science, especially those interested in set theory

and its relationship to logic." --Computing Reviews

"Intended as a text for advanced undergraduates and graduate students. Essentially self-contained."--The Bulletin of Mathematics Books

"The book under review is a textbook for a beginning graduate course on set theory. The structure is fairly standard, with the book divided into three main sections; after an introductory section developing the basic facts about the universe of set theory, there is a section on constructibility and

a section on forcing. The main goals of the book are to give proofs that the axiom ofchoice (AC) and the generalised continuum hypothesis (GCH) are consistent with and independent of the axioms of Zermelo-Fraenkel set theory (ZF). . . . The distinctive features of this book are the use of class set

theory, the treatment of induction, and the use of modal logic in the treatment of forcing. The writing is lucid and accurate, and the main theorems are proved in an efficient way."--Journal of Symbolic Logic "Smullyan and Fitting. . .achieve miraculous clarity in a subject crowded with intimidating espositions; in particular their book meets the very high standard of exposition set by Smullyan's previous works." --Choice

"This text is a general introduction to NBG (von Neumann-Bernays-Godel class-set theory), and to Godel and Cohen proofs of the relative consistency and the independence of the generalized continuum hypothesis (GCH) and the axiom of choice (AC). . . .The authors write with admirable lucidity. There

are some truly charming set pieces on countability and uncountability and on mathematical induction--I intend to appropriate them for my classes. . . .this is an excellent book for anyone interested in set theory and foundations."--Mathematical Reviews

"A well-written discussion of set theory, and readers will need a solid background in mathematics to fully appreciate its contents. The book is self-contained and intended for advanced undergraduates and graduate students in mathematics and computer science, especially those interested in set theory

and its relationship to logic." --Computing Reviews

"Intended as a text for advanced undergraduates and graduate students. Essentially self-contained."--The Bulletin of Mathematics Books

"The book under review is a textbook for a beginning graduate course on set theory. The structure is fairly standard, with the book divided into three main sections; after an introductory section developing the basic facts about the universe of set theory, there is a section on constructibility and

a section on forcing. The main goals of the book are to give proofs that the axiom of choice (AC) and the generalised continuumhypothesis (GCH) are consistent with and independent of the axioms of Zermelo-Fraenkel set theory (ZF). . . . The distinctive features of this book are the use of class set

theory, the treatment of induction, and the use of modal logic in the treatment of forcing. The writing is lucid and accurate, and the main theorems are proved in an efficient way."--Journal of Symbolic Logic

show more