Axiomatic Set Theory (eBook)

Front Cover 1
Axiomatic Set Theory: Impredicative Theories of Classes 4
Copyright Page 5
Table of Contents 12
PART 1: INTRODUCTION 18
Chapter 1.1 Informal background 20
Chapter 1.2 Axioms 22
1.2.1 General axioms 22
1.2.2 Limiting axioms 25
PART 2: GENERAL CLASS THEORY 28
Chapter 2.1 Introduction to General Class Theory 30
2.1.1 Axioms for G 30
2.1.2 Ordered pairs and Cartesian product 40
2.1.3 Generalized operations 43
(+) 2.1.4 G is a subtheory of B 48
Chapter 2.2 Relations 50
2.2.1 Operations with relations 50
2.2.2 Relations as superclasses 57
2.2.3 Types of relations 59
Chapter 2.3 Functions and Operations 68
2.3.1 Functional relations 68
2.3.2 Monotone operations 74
2.3.3 Additive and multiplicative operations 78
Chapter 2.4 Natural numbers 84
2.4.1 Fundamental properties 84
2.4.2 Arithmetic of natural numbers 90
Chapter 2.5 Generalized Recursive Definitions 97
2.5.1 The ancestry relation 97
2.5.2 Recursion over well-founded relations 103
2.5.3 Well-founded classes 107
Chapter 2.6 Well-orderings 112
2.6.1 Isomorphisms and simple orderings 112
2.6.2 Well - ordering relations 115
Chapter 2.7 Cardinality 122
2.7.1 Equipollency 122
2.7.2 Cardinal addition and multiplication 125
2.7.3 Finite and denumerable classes 127
2.7.4 Finite and denumerable orderings 127
PART 3: MORSE-KELLEY-TARSKI CLASS THEORY 138
Chapter 3.1 Introduction to Morse-Kelley-Tarski Class Theory 140
3.1.1 Axiomatic system 140
3.1.2 Elementary consequences 143
3.1.3 Relations and functions 146
(+) 3.1.4 MKT is a subtheory of B 147
Chapter 3.2 Structure of well-founded classes 155
3.2.1 Well -founded classes in MKT 155
3.2.2 Rank 160
3.2.3 Universal domains with a given rank 167
Chapter 3.3 Ordinal Numbers 173
3.3.1 General properties of ordinals 173
3.3.2 Ordinal functions 180
3.3.3 Iteration of functions 186
3.3.4 Ordinal addition 190
3.3.5 Ordinal multiplication 196
3.3.6 Ordinal exponentiation 201
Chapter 3.4 Cardinal Numbers 209
3.4.1 Definition of cardinal numbers 209
3.4.2 Cardinal exponentiation 211
3.4.3 Finite multiples of cardinals and countable sums 216
3.4.4 Bradford's result and its consequences 230
Chapter 3.5 Simple Orderings 274
3.5.1 Types of simple orderings 274
3.5.2 Scattered, dense, and continuous orderings 282
3.5.3 Well -Orderings 290
Chapter 3.6 Alephs 293
3.6.1 Arithmetic of alephs 293
3.6.2 Hartog's function 299
Chapter 3.7 The Local Axiom of Choice and the Generalized Continuum Hypothesis 305
3.7.1 Forms of the local axiom of choice 305
3.7.2 Maximal principles 308
3.7.3 Cardinal equivalences 309
3.7.4 The generalized continuum hypothesis implies AC 312
Chapter 3.8 Gauging Sizes of Sets and Classes 314
3.8.1 Initial ordinals 314
3.8.2 Degree of cofinality 317
3.8.3 The exponential scale 325
3.8.4 Inaccessible sets 328
3.8.5 Closed unbounded and stationary classes 330
PART 4: MORSE-KELLEY-TARSKI CLASS THEORY WITH CHOICE 336
Chapter 4.1 The Global Axiom of Choice 338
4.1.1 Axioms of choice 338
4.1.2 Immediate consequences 339
Chapter 4.2 Cardinality Theory 342
4.2.1 Initial ordinals as cardinals 342
4.2.2 Infinite sums and products 344
4.2.3 Degree of cofinality with choice 347
4.2.4 Cardinal exponentiation of initial ordinals 349
4.2.5 The exponential scale of initial ordinals 352
Chapter 4.3 Filters , Ideals, and Trees 357
4.3.1 Filters and ideals 357
4.3.2 The closed unbounded filter and the thin ideal 361
4.3.3 Trees 363
PART 5: BERNAYS CLASS THEORY 368
Chapter 5.1 Inaccessibility 370
5.1.1 Inaccessible sets 370
5.1.2 Inaccessible cardinals 373
5.1.3 Classification of inaccessible cardinals 376
Chapter 5.2 Weakly Compact and Larger Cardinals 383
5.2.1 Weakly compact cardinals 383
5.2.2 Larger cardinals 392
References 394
Index of Symbols 398