Holdings: Set theory and the continuum problem

Set theory and the continuum problem:

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Bibliographic Details
Main Author:	Smullyan, Raymond M. 1919-2017 (Author)
Format:	Book
Language:	English
Published:	Mineola, N.Y. Dover Publ. 2010
Edition:	Rev. and corr. republ.
Subjects:	Set theory Continuum hypothesis Kontinuumshypothese Mengenlehre
Online Access:	Publisher description Inhaltsverzeichnis Klappentext
Item Description:	Includes bibliographical references and index
Physical Description:	XIII, 315 S.
ISBN:	9780486474847 0486474844

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MARC


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adam_text	CONTENTS PREFACE TO THE REVISED 2010 EDITION iii PREFACE v I AXIOMATIC SET THEORY 1 CHAPTER 1 GENERAL BACKGROUND 3 §1 What is infinity? .................................... 3 §2 Countable or uncountable? .............................. 4 §3 A non-denumerable set ................................ 6 §4 Larger and smaller .................................. 7 §5 The continuum problem ................................ 9 §6 Significance of the results ............................... 9 §7 Frege set theory .................................... 11 §8 Russell s paradox ................................... 11 §9 Zermelo set theory .................................. 12 §10 Sets and classes .................................... 13 CHAPTER 2 SOME BASICS OF CLASS-SET THEORY 15 §1 Extensionality and separation ............................. 15 §2 Transitivity and supercompleteness .......................... 17 §3 Axiom of the empty set ................................ 18 §4 The pairing axiom ................................... 19 §5 The union axiom ................................... 21 §6 The power axiom ................................... 23 §7 Cartesian products .................................. 24 §8 Relations ....................................... 24 §9 Functions ....................................... 25 §10 Some useful facts about transitivity .......................... 26 §11 Basic universes .................................... 27 CHAPTERS THE NATURAL NUMBERS 29 §1 Preliminaries ..................................... 29 §2 Definition of the natural numbers ........................... 32 §3 Derivation of the Peano postulates and other results ................. 33 §4 A double induction principle and its applications .................. 35 §5 Applications to natural numbers ........................... 40 §6 Finite sets ....................................... 42 §7 Denumerable classes ................................. 42 §8 Definition by finite recursion ............................. 43 CONTENTS §9 Supplement—optional................................ 44 CHAPTER 4 SUPERINDUCTION, WELL ORDERING AND CHOICE 47 §1 Introduction to well ordering............................. 47 §2 Superinduction and double superinduction ...................... 52 §3 The well ordering of ^-towers ............................ 56 §4 Well ordering and choice ............................... 57 §5 Maximal principles .................................. 61 §6 Another approach to maximal principles ....................... 64 §7 Cowen s theorem ................................... 66 §8 Another characterization of g-sets .......................... 68 CHAPTERS ORDINAL NUMBERS 71 §1 Ordinal numbers ................................... 71 §2 Ordinals and transitivity ................................ 75 §3 Some ordinals ..................................... 76 CHAPTER 6 ORDER ISOMORPHISM AND TRANSFINITE RECURSION 79 §1 A few preliminaries .................................. 79 §2 Isomorphisms of well orderings ........................... 80 §3 The axiom of substitution ............................... 82 §4 The counting theorem ................................. 83 §5 Transfinite recursion theorems ............................ 84 §6 Ordinal arithmetic ................................... 88 CHAPTER? RANK 91 §1 The notion of rank .................................. 91 §2 Ordinal hierarchies .................................. 92 §3 Applications to the Ra sequence ........................... 93 §4 Zermelo universes ................................... 96 CHAPTER 8 FOUNDATION, є -INDUCTION, AND RANK 99 §1 The notion of well-foundedness ........................... 99 §2 Descending є -chains ................................. 100 §3 e-Induction and rank ................................. 101 §4 Axiom £ and Von Neumann s principle ....................... 103 §5 Some other characterizations of ordinals ....................... 104 §6 More on the axiom of substitution .......................... 106 CHAPTER 9 CARDINALS 107 § 1 Some simple facts ................................... 107 §2 The Bernstein-Schröder theorem ........................... 108 §3 Denumerable sets ................................... Ill §4 Infinite sets and choice functions ........................... Ill §5 Hartog s theorem ................................... 112 §6 A fundamental theorem ................................ 114 §7 Preliminaries ..................................... 116 §8 Cardinal arithmetic .................................. 118 §9 Sierpinski s theorem ................................. 121 CONTENTS xi II CONSISTENCY OF THE CONTINUUM HYPOTHESIS 125 CHAPTER 10 MOSTOWSKI-SHEPHERDSON MAPPINGS 127 §1 Relational systems .................................. 127 §2 Generalized induction and Г-гапк .......................... 129 §3 Generalized transfinite recursion ........................... 133 §4 Mostowski-Shepherdson maps ............................ 134 §5 More on Mostowski-Shepherdson mappings ..................... 136 §6 Isomorphisms, Mostowski-Shepherdson, well orderings ............... 137 CHAPTER 11 REFLECTION PRINCIPLES 141 §0 Preliminaries ..................................... 141 §1 The Tarski-Vaught theorem .............................. 145 §2 We add extensionality considerations ......................... 147 §3 The class version of the Tarski-Vaught theorems ................... 148 §4 Mostowski, Shepherdson, Tarski, and Vaught .................... 150 §5 The Montague-Levy reflection theorem ....................... 151 CHAPTERS CONSTRUCTIBLE SETS 155 §0 More on first-order definability ............................ 155 §1 The class L of constructible sets ........................... 156 §2 Absoluteness ..................................... 158 §3 Constructible classes ................................. 163 CHAPTER 13 L IS A WELL FOUNDED FIRST-ORDER UNIVERSE 169 §1 First-order universes ................................. 169 §2 Some preliminary theorems about first-order universes ............... 172 §3 More on first-order universes ............................. 174 §4 Another result ..................................... 177 CHAPTER 14 CONSTRUCITBILITY IS ABSOLUTE OVER L 179 §1 Σ -formulas and upward absoluteness ......................... 179 §2 More on Σ definability ................................ 181 §3 The relation y = T(x) ................................. 183 §4 Constructibffity is absolute over L .......................... 189 §5 Further results ..................................... 190 §6 A proof that ¿can be well ordered .......................... 191 CHAPTER 15 CONSTRUCTIBILITY AND THE CONTINUUM HYPOTHESIS 193 §0 What we will do .................................... 193 §1 The key result ..................................... 194 §2 Gödel s isomorphism theorem (optional) ....................... 196 §3 Some consequences of Theorem G .......................... 198 §4 Metamaíhematical consequences of Theorem G ................... 199 §5 Relative consistency of the axiom of choice ..................... 200 §6 Relative consistency ofGCH and AC in class-set theory ............... 201 CONTENTS Ш FORCING AND INDEPENDENCE RESULTS 205 CHAPTER 16 FORCING, THE VERY IDEA 207 §1 What is forcing? ....................................207 §2 What is modal logic? .................................209 §3 What is 54 and why do we care? ...........................213 §4 A classical embedding ................................215 §5 The basic idea .....................................221 CHAPTER 17 THE CONSTRUCTION OF 54 MODELS FOR ZF 223 §1 What are the models? .................................223 §2 About equality ....................................229 §3 The well founded sets are present ...........................233 §4 Four more axioms ...................................235 §5 The definability of forcing ..............................239 §6 The substitution axiom schema ............................242 §7 The axiom of choice .................................244 §8 Where we stand now .................................247 CHAPTER 18 THE AXIOM OF CONSTRUCTIBILITY IS INDEPENDENT 249 §1 Introduction ...................................... 249 §2 Ordinals are well behaved ............................... 249 §3 Constractible sets are well behaved too ........................ 251 §4 A real 54 model, at last ................................ 252 §5 Cardinals are sometimes well behaved ........................ 253 §6 The status of the generalized continuum hypothesis ................. 256 CHAPTERS INDEPENDENCE OF THE CONTINUUM HYPOTHESIS 259 §1 Power politics .....................................259 §2 The model .......................................259 §3 Cardinals stay cardinals ................................260 §4 CH is independent ...................................262 §5 Cleaning it up .....................................263 §6 Wrapping it up ....................................266 CHAPTER 20 INDEPENDENCE OF THE AXIOM OF CHOICE 267 §1 A little history ..................................... 267 §2 Automorphism groups ................................ 268 §3 Automorphisms preserve truth ............................ 270 §4 Model and submodel ................................. 272 §5 Verifying the axioms ................................. 273 §6 AC is independent ................................... 278 CHAPTER 21 CONSTRUCTING CLASSICAL MODELS 285 §1 On countable models ................................. 285 §2 Cohen s way ...................................... 286 §3 Dense sets, filters, and generic sets .......................... 287 §4 When generic sets exist ................................ 289 §5 Generic extensions .................................. 291 §6 The truth lemma .................................... 294 CONTENTS xiii §7 Conclusion ......................................296 CHAPTER 22 FORCING BACKGROUND 299 §1 Introduction ...................................... 299 §2 Cohen s version^) .................................. 300 §3 Boolean valued models ................................ 301 §4 Unramifled forcing .................................. 301 §5 Extensions ....................................... 302 BIBLIOGRAPHY 305 INDEX 307 LIST OF NOTATION ЗІЗ Set Theory and the Continuum Problem umilili ill SiìiiSf ii lililí! Fittili A lucid, elegant, and complete survey of set theory, this volume is drawn from the authors substantial teaching experience. The first of three parts focuses on axiomatic set theory. The second part explores the consistency of the continuum hypothesis, and the final section examines forcing and independence results. Part One s focus on axiomatic set theory features nine chapters that examine problems related to size comparisons between infinite sets, basics of class the¬ ory, and natural numbers. Additional topics include author Raymond Smullyan s double induction principle, super induction, ordinal numbers, order isomorphism and transfinite recursion, and the axiom of foundation and cardinals. The six chapters of Part Two address Mostowski-Shepherdson mappings, reflection prin¬ ciples, constructible sets and constructibiiity, and the continuum hypothesis. The text concludes with a seven-chapter exploration of forcing and independence results. This treatment is noteworthy for its clear explanations of highly techni¬ cal proofs and its discussions of countability, uncountability, and mathematical induction, which are simultaneously charming for experts and understandable to graduate students of mathematics. Dover (2010) revised and updated republication of the edition published by Oxford University Press, New York, 1996.
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spelling	Smullyan, Raymond M. 1919-2017 Verfasser (DE-588)115668411 aut Set theory and the continuum problem Raymond M. Smullyan ; Melvin Fitting Rev. and corr. republ. Mineola, N.Y. Dover Publ. 2010 XIII, 315 S. txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Set theory Continuum hypothesis Kontinuumshypothese (DE-588)4481570-0 gnd rswk-swf Mengenlehre (DE-588)4074715-3 gnd rswk-swf Mengenlehre (DE-588)4074715-3 s Kontinuumshypothese (DE-588)4481570-0 s DE-604 Fitting, Melvin Sonstige oth http://www.loc.gov/catdir/enhancements/fy1003/2009036172-d.html Publisher description Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020470107&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020470107&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext
spellingShingle	Smullyan, Raymond M. 1919-2017 Set theory and the continuum problem Set theory Continuum hypothesis Kontinuumshypothese (DE-588)4481570-0 gnd Mengenlehre (DE-588)4074715-3 gnd
subject_GND	(DE-588)4481570-0 (DE-588)4074715-3
title	Set theory and the continuum problem
title_auth	Set theory and the continuum problem
title_exact_search	Set theory and the continuum problem
title_full	Set theory and the continuum problem Raymond M. Smullyan ; Melvin Fitting
title_fullStr	Set theory and the continuum problem Raymond M. Smullyan ; Melvin Fitting
title_full_unstemmed	Set theory and the continuum problem Raymond M. Smullyan ; Melvin Fitting
title_short	Set theory and the continuum problem
title_sort	set theory and the continuum problem
topic	Set theory Continuum hypothesis Kontinuumshypothese (DE-588)4481570-0 gnd Mengenlehre (DE-588)4074715-3 gnd
topic_facet	Set theory Continuum hypothesis Kontinuumshypothese Mengenlehre
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