Volume II, on formal (ZFC) set theory, incorporates a self-contained "chapter 0" on proof techniques so that it is based on formal logic, in the style of Bourbaki. The emphasis on basic techniques provides a solid foundation in set theory and a thorough context for the presentation of advanced topics (such as absoluteness, relative consistency results, two expositions of Godel's construstive universe, numerous ways of viewing recursion and Cohen forcing).
Author(s): George Tourlakis
Series: Cambridge Studies in Advanced Mathematics 83
Publisher: Cambridge University Press
Year: 2003
Language: English
Pages: 575
Cover......Page 1
Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 13
I A Bit of Logic: A User’s Toolbox......Page 19
I.1. First Order Languages......Page 25
Logical Symbols.......Page 27
Nonlogical Symbols.......Page 28
I.2. A Digression into the Metatheory: Informal Induction and Recursion......Page 38
I.3. Axioms and Rules of Inference......Page 47
I.4. Basic Metatheorems......Page 61
I.5. Semantics......Page 71
I.6. Defined Symbols......Page 84
I.7. Formalizing Interpretations......Page 95
I.8. The Incompleteness Theorems......Page 105
I.9. Exercises......Page 112
II.1. The “Real Sets”......Page 117
II.2. A Naïve Look at Russell’s Paradox......Page 123
II.3. The Language of Axiomatic Set Theory......Page 124
II.4. On Names......Page 128
III.1. Extensionality......Page 132
III.2. Set Terms; Comprehension; Separation......Page 137
III.3. The Set of All Urelements; the Empty Set......Page 148
III.4. Class Terms and Classes......Page 152
III.5. Axiom of Pairing......Page 163
III.6. Axiom of Union......Page 167
III.7. Axiom of Foundation......Page 174
III.8. Axiom of Collection......Page 178
III.9. Axiom of Power Set......Page 196
III.10. Pairing Functions and Products......Page 200
III.11. Relations and Functions......Page 211
III.12. Exercises......Page 228
IV.1. Introduction......Page 233
IV.2. More Justification for AC; the “Constructible” Universe Viewpoint......Page 236
IV.3. Exercises......Page 247
V.1. The Natural Numbers......Page 250
V.2. Algebra of Relations; Transitive Closure......Page 271
V.3. Algebra of Functions......Page 290
V.4. Equivalence Relations......Page 294
V.5. Exercises......Page 299
VI.1. PO Classes, LO Classes, and WO Classes......Page 302
VI.2. Induction and Inductive Definitions......Page 311
VI.3. Comparing Orders......Page 334
VI.4. Ordinals......Page 341
VI.5. The Transfinite Sequence of Ordinals......Page 358
VI.6. The von Neumann Universe......Page 376
VI.7. A Pairing Function on the Ordinals......Page 391
VI.8. Absoluteness......Page 395
VI.9. The Constructible Universe......Page 413
VI.10. Arithmetic on the Ordinals......Page 428
VI.11. Exercises......Page 444
VII Cardinality......Page 448
VII.1. Finite vs. Infinite......Page 449
VII.2. Enumerable Sets......Page 460
VII.3. Diagonalization; Uncountable Sets......Page 469
VII.4. Cardinals......Page 475
VII.5. Arithmetic on Cardinals......Page 488
VII.6. Cofinality; More Cardinal Arithmetic; Inaccessible Cardinals......Page 496
VII.7. Inductively Defined Sets Revisited; Relative Consistency of GCH......Page 512
VII.8. Exercises......Page 530
VIII Forcing......Page 536
VIII.1. PO Sets, Filters, and Generic Sets......Page 538
VIII.2. Constructing Generic Extensions......Page 542
VIII.3. Weak Forcing......Page 546
VIII.4. Strong Forcing......Page 550
VIII.5. Strong vs. Weak Forcing......Page 561
VIII.6. M[G] Is a CTM of ZFC If M Is......Page 562
VIII.7. Applications......Page 567
VIII.8. Exercises......Page 576
Bibliography......Page 578
List of Symbols......Page 581
Index......Page 585
