Set Theory: An Introduction to Axiomatic Reasoning

Preface

I   Axioms and classes

    1 Classes, sets and axioms

    2 Constructing classes and sets

II   Class operations

    3 Operations on classes and sets

    4 Cartesian products

III   Relations

    5 Relations on a class or set

    6 Equivalence relations and order relations

    7 Partitions induced by equivalence relations

    8 Equivalence classes and quotient sets

IV   Functions

    9 Functions: A set-theoretic definition

    10 Operations on functions

    11 Images and preimages of sets

    12 Equivalence relations induced by functions

V   From sets to numbers

    13 Natural numbers

    14 The natural numbers as a well-ordered set

    15 Arithmetic of the natural numbers

    16 The integers Z and the rationals Q

    17 Real numbers: “Dedekind cuts are us!”

VI   Infinite sets

    18 Infinite sets versus finite sets

    19 Countable and uncountable sets

    20 Equipotence as an equivalence relation

    21 The Schröder-Bernstein theorem

VII   Cardinal numbers

    22 Introduction to cardinal numbers

    23 Addition and multiplication in C

    24 Exponentiation of cardinal numbers

    25 On sets of cardinality c

VIII   Ordinal numbers

    26 More on well-ordered sets

    27 Ordinals: definition and properties

    28 Properties of the class of ordinals.

    29 Cardinal numbers: “Initial ordinals are us!”

IX   Choice, regularity and Martin's axiom

    30 Axiom of choice

    31 Axiom of regularity

    32 Cumulative hierarchy

    33 Martin's axiom

X   Ordinal arithmetic

    34 Ordinal addition

    35 Ordinal multiplication and exponentiation

XI   Appendix

    Appendix A: Boolean algebras and Martin's axiom

Bibliography

Index