INSTYTUT MATEMATYKI
Uniwersytet Śląski
40-007 Katowice, ul. Bankowa 14


tel/fax (032) 2582976
e-mail:im@ux2.math.us.edu.pl


Contents

Introduction  IX

Index of Symbols XIII

Basic Set Theory


1. Sets,relations,functions
1.1. Sets 3
1.2 Relations and functions 8
1.3 Equivalence relations 16
Commentaries 18

2. Orderings
2.1. Partially ordered and linearly ordered sets 20
2.2 Kuratowski—Zorn Lemma 28
2.3 Well-Ordered sets.  Zermelo’s Theorem 31
2.4 Lexicographic order 33
2.5 Dedekind cuts 35
Commentaries 38

3. Natural and rational numbers
3.1. Natural numbers 40
3.2 Recursive definitions 42
3.3 Operations on natural numbers 46
3.4 Rational non-negative numbers 52
Commentaries 56
4. Field of the real numbers
4.1. Non-negative real numbers 58
4.2 Operations on non-negative real numbers 59
4.3 A construction of the set of real numbers 64
5. Equinumerosity
5.1. Equinumerous sets 67
5.2 Finite sets 72
5.3 Countable sets 80
5.4 Uncountable sets 84
5.5 Order characterization of real numbers 87
Commentaries 91

Axiomatic Set Theory

6. Axioms
6.1. Set theory as a first order theory 97
6.2 Axioms of the ZFC set theory 99
Commentaries 105

7. Ordinal numbers
7.1. Transitive sets and ordinal numbers 107
7.2 The number ω  and finite ordinal numbers 112
7.3 Order types of well ordered sets 115
7.4 Transfinite  recursion and hierarchy of sets 118
7.5 Ordinal  arithmetic 126
Commentaries 139
8. Cardinal numbers
8.1. Cardinal number of a set 141
8.2 Addition  and multiplication of cardinals 145
8.3 Regular and singular cardinals 150
8.4 Exponentiation of  cardinals 153
Commentaries 159
9. Combinatorial properties of sets
9.1. Almost disjoint sets, Hausdorff’s gaps 163
9.2 Partition relations, Ramsey Theorem 172
9.3 Stationary sets,  Erdös-Rado Theorem 180
9.4 Martin’s Axiom 187
Commentaries 192

Classical Constructions of Set Theory

10. Lattices
10.1. Distributive lattices  201
10.2 Homomorphisms of lattices  206
10.3 Filters and ideals  211
10.4 Boolean lattices  215
Commentaries  220
11. Topologies
11.1. Topological spaces  227
11.2 Metric spaces  243
11.3 Compact spaces  261
11.4 Products and cubes  271
11.5 Stone spaces  281
Commentaries  290
12. Trees
12.1. Trees and linearly ordered sets  286
12.2 Aronszajn tree  300
12.3 Suslin tree  304
Commentaries  308
13. Measures
13.1. Lebesgue  measure on the real line    311
13.2 Measures on σ-fields   324
13.3 Measures on cardinals   329
Commentaries   336
14. Boolean algebras
14.1 Representations of Boolean algebras 342
14.2 Complete Boolean algebras, completions 348
14.3 Free Boolean algebras, independent sets 352
14.4 Quotient Boolean algebras 358
14.4.1 Boolean algebra of all subsets of ω modulo ideal of finite sets 359
14.4.2 Boolean algebra of Borel sets modulo ideal of first category sets 361
14.4.3 Boolean algebra of Borel sets modulo ideal of null sets 363
14.5 Measures on Boolean algebras 365
Commentaries 368
15. Ramsey theory
15.1. Compact semigroups   373
15.2 Hindman’s Theorem   376
15.3 Hales-Jewett and van der Waerden’s Theorems   380
Commentaries   387

Around the Axiom of Choice

16. Equivalent versions of the Axiom of Choice
16.1. Multiple choice 393
16.2 Bases in linear spaces 395
16.3 Tychonoff’s Product Theorem 398
16.4 Prime  Ideal Theorem in lattices 398
Commentaries 399
17. Weaker versions of the Axiom of Choice
17.1. Prime Ideal Theorem in Boolean algebras 402
17.2 Dependent Choice versus countable Axiom of   Choice 408
17.3 Axiom of Determinancy 415
Commentaries 427
18. Banach-Tarski Paradox
18.1. Paradoxical set 429
18.2 Banach – Tarski Theorem 435
Commentaries 439

       Bibliography      441
     Subject index      443
     Index of Authors      449
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