Set Theory, with an Introduction to Descriptive Set Theory (Studies in Logic and the Foundations of Mathematics - Vol 86)

PREFACE TO THE FIRST EDITION PREFACE TO THE SECOND EDITION CONTENTS CHAPTER I: ALGEBRA OF SETS § 1. Propositional calculus § 2. Sets and operations on sets § 3. Inclusion. Empty set § 4. Laws of union, intersection, and subtraction § 5. Properties of symmetric difference § 6. The set 1, complement § 7. Constituents § 8. Applications of the algebra of sets to topology § 9. Boolean algebras § 10. Lattices CHAPTER II: AXIOMS OF SET THEORY. RELATIONS. FUNCTIONS § 1. Set-theoretical formulas. Quantifiers § 2. Axioms of set theory § 3. Some simple consequences of the axioms § 4. Cartesian products. Relations § 5. Equivalence relations. Partitions § 6. Functions § 7. Images and inverse images § 8. Functions consistent with a given equivalence relation. Factor Boolean algebras § 9. Order relations § 10. Relational systems, their isomorphisms and types CHAPTER III: NATURAL NUMBERS. FINITE AND INFINITE SETS § 1. Natural numbers § 2. Definitions by induction § 3. The mapping J of the set NxN onto N and related mappings § 4. Finite and infinite sets CHAPTER IV: GENERALIZED UNION, INTERSECTION ANDCARTESIAN PRODUCT § 1. Set-valued functions. Generalized union and intersection § 2. Operations on infinite sequences of sets § 3. Families of sets closed under given operations § 4. sigma-additive and delta-muItiplicative families of sets § 5. Reduction and separation properties § 6. Generalized cartesian products § 7. Cartesian products of topological spaces § 8. The Tychonoff theorem § 9. Reduced direct products § 10. Infinite operations in lattices and in Boolean algebras § 11. Extensions of ordered sets to complete lattices § 12. Representation theory for distributive lattices CHAPTER V: THEORY OF CARDINAL NUMBERS § 1. Equipotence. Cardinal numbers § 2. Countable sets § 3. The hierarchy of cardinal numbers § 4. The arithmetic of cardinal numbers § 5. Inequalities between cardinal numbers. The Cantor-Bernstein theorem and its generalizations § 6. Properties of the cardinals a and c § 7. The generalized sum of cardinal numbers § 8. The generalized product of cardinal numbers CHAPTER VI: LINEARLY ORDERED SETS § 1. Introduction § 2. Dense, scattered, and continuous sets § 3. Order types omega, eta, and lambda § 4. Arithmetic of order types § 5. Lexicographical ordering CHAPTER VII: WELL-ORDERED SETS § 1. Definitions. Principle of transfinite induction § 2. Ordinal numbers § 3. Transfinite sequences § 4. Definitions by transfinite induction § 5. Ordinal arithmetic § 6. Ordinal exponentiation § 7. Expansions of ordinal numbers for an arbitrary base § 8. The well-ordering theorem § 9. Von Neumann's method of elimination of ordinal numbers CHAPTER VIII: ALEPHS AND RELATED TOPICS § 1. Ordinal numbers of power a § 2. The cardinal Aleph(m). Hartogs' aleph § 3. Initial ordinals § 4. Alephs and their arithmetic § 5. The exponentiation of alephs § 6. The exponential hierarchy of cardinal numbers § 7. The continuum hypothesis § 8. The number of prime ideals in the algebra p(A) § 9. m-disjoint sets § 10. Families of disjoint open sets § 11. Equivalence of certain statements about cardinal numbers with the axiom of choice CHAPTER IX: TREES AND PARTITIONS § 1. Trees § 2. The lexicographical ordering of zero-one sequences. eta_xi sets § 3. Koenig's infinity lemma § 4. Aronszajn's trees § 5. Souslin trees § 6. Some partition theorems CHAPTER X: INACCESSIBLE CARDINALS § 1. Normal functions and stationary sets § 2. Weakly and strongly inaccessible cardinals § 3. A digression on models of Sigma0[TR] § 4. Higher types of inaccessible numbers § 5. Weakly compact cardinals § 6. Measurable cardinals § 7. Measurable cardinals and reduced products CHAPTER XI: AUXILIARY NOTIONS § 1. The notion of a metric space. Various fundamental topological notions § 2. Exponential topology. Compact-open topology § 3. Complete and Polish spaces § 4. L-measurable mappings § 5. The operation A (see Lusin and Souslin [1]) § 6. The Lusin sieve (see Lusin [4]) INTRODUCTION TO DESCRIPTIVE SET THEORY CHAPTER XII: BOREL SETS. B-MEASURABLE FUNCTIONS. BAIRE PROPERTY § 1. Elementary properties of Borel subsets of a metric space § 2. Ambiguous Borel sets § 3. Borel-measurable functions § 4. B-measurable complex and product functions § 5. Universal functions for Borel classes § 6. Borel subsets of Polish spaces § 7. Further properties of Borel sets § 8. Baire property CHAPTER XIII: SOUSLIN SPACES. PROJECTIVE SETS § 1. Souslin spaces. Fundamental properties § 2. Applications of countable order types to Souslin spaces § 3. Coanalytic sets (CA-sets) § 4. The a-a1gebra S generated by Souslin sets and the S-measurable mappings § 5. The PCA-sets and sets of higher projective classes CHAPTER XIV: MEASURABLE SELECTORS § 1. The general selector theorem § 2. Selectors for measurable partitions of Polish spaces § 3. Selectors for point-inverses of continuous mappings LIST OF IMPORTANT SYMBOLS SUBJECT INDEX BIBLIOGRAPHY