Set theory can be rigorously and profitably studied through an intuitive approach, thus independently of formal logic. Nearly every branch of Mathematics depends upon set theory, and thus, knowledge of set theory is of interest to every mathematician. This book is addressed to all mathematicians and tries to convince them that this intuitive approach to axiomatic set theory is not only possible but also valuable.
The book has two parts. The first one presents, from the sole intuition of "collection" and "object", the axiomatic ZFC-theory. Then, we present the basics of the theory: the axioms, well-orderings, ordinals and cardinals are the main subjects of this part. In all, one could say that we give some standard interpretation of set theory, but this standard interpretation results in a multiplicity of universes.
The second part of the book deals with the independence proofs of the continuum hypothesis (CH) and the axiom of choice (AC), and forcing is introduced as a necessary tool, and again the theory is developed intuitively, without the use of formal logic. The independence results belong to the metatheory, as they refer to things that cannot be proved, but the greater part of the arguments leading to the independence results, including forcing, are purely set-theoretic.
The book is self-contained and accessible to beginners in set theory. There are no prerequisites other than some knowledge of elementary mathematics. Full detailed proofs are given for all the results.
I. The Zermelo-Fraenkel Theory
1. Introduction
1.1 The beginnings of set theory
1.2 The antinomies
1.3 Axiomatic theories
1.4 Intuitive and formal axiomatic theories
1.5 Axiomatic set theory
1.6 Logical symbolism and truth tables
1.7 Permutations
2. Objects, Collections, Sets
2.1 Introduction
2.2 Membership and inclusion
2.3 Intersections and unions
2.4 Differences
2.5 The first axioms
2.6 Some remarks on the natural numbers
2.7 Cartesian products
2.8 Exercises
3. Classes
3.1 The formation of classes
3.1.1 Class sequences
3.1.2 Obtaining new classes
3.2 Classes and formulas
3.2.1 Formulas describe classes
3.2.2 The tree of a formula
3.2.3 Other operations with classes
4. Relations
4.1 Relations and operations with relations
4.2 Order relations
4.2.1 Some types of relations
4.2.2 Order relations
4.2.3 Special elements in ordered classes
4.3 Functional relations
4.3.1 Definitions and notations
4.3.2 Inversion and composition of functionals
4.4 Partitions and equivalence relations
4.5 Exercises
5. Maps, Orderings, Equivalences
5.1 Central axioms and separation
5.1.1 The axioms
5.1.2 The principle of separation
5.1.3 Some consequences of the axioms
5.2 Maps
5.2.1 Functions and maps
5.2.2 Injective and surjective maps
5.2.3 Relations, partitions, operations
5.3 Exercises
6. Numbers and Infinity
6.1 The axiom of infinity
6.1.1 Dedekind-Peano sets
6.1.2 Inductive sets
6.2 A digression on the axiom of infinity
6.2.1 The set ω and natural numbers
6.2.2 Axiom 6 and property NNS
6.3 The principle of induction
6.3.1 Natural induction
6.3.2 Well-orderings
6.3.3 Other forms of induction
6.4 The recursion theorems
6.5 The operations in ω.
6.6 Countable sets
6.7 Integers and rationals
6.7.1 The ring Z of integers
6.7.2 The field of rational numbers
6.8 The field of real numbers
6.9 Exercises
7. Pure Sets
7.1 Transitivity
7.1.1 Transitive closure of a set
7.1.2 Pure sets
7.1.3 Classes as universes
7.2 ZF--universes
7.2.1 The universe of pure sets
7.2.2 The axiom of extension
7.3 Relativization of classes
7.3.1 Extensional apt classes
7.3.2 Relativization
7.3.3 C-absolute classes
7.4 Exercises
8. Ordinals
8.1 Well ordered classes and sets
8.2 Morphisms between ordered sets
8.3 Ordinals
8.4 Induction and recursion
8.5 Ordinal arithmetic
8.5.1 Ordinal addition
8.5.2 Ordinal multiplicatio
8.6 Exercises
9. ZF-Universes
9.1 Well founded sets
9.2 The von Neumann universe of a universe
9.2.1 The construction of V.
9.2.2 V and the axioms
9.2.3 The axiom of regularity
9.2.4 Absoluteness in ZF.universes
9.3 Well founded relations
9.3.1 Induction and recursion for well founded relations
9.3.2 Mostowski's theorem
9.4 Exercises
10. Cardinals and the Axiom of Choice
10.1 Equipotent sets and cardinals
10.2 Operations with cardinals
10.2.1 Addition
10.2.2 Multiplication
10.3 The axiom of choice
10.3.1 Choice and cardinals of sets
10.3.2 Equivalent forms of the axiom of choice
10.4 Finite and infinite sets
10.4.1 Finiteness criteria
10.4.2 The series of the alephs
10.5 Cardinal exponentiation
10.5.1 Exponentiation and the continuum hypothesis
10.5.2 Infinite sums and products
10.6 Cofinalities
10.6.1 The cofinality of an ordinal
10.6.2 More on cardinal arithmetic
10.7 Exercises
II. Independence Results
11. Countable Universes
11.1 The metatheory of sets
11.2 Extensions and reliability
11.3 ZF(C)*-universes
11.4 Reflection theorems
11.5 Inaccessible cardinals
11.6 Exercises
12. The constructible universe
12.1 X-conxtructible sets
12.2 The constructible hierarchy
12.3 The constructible universe
12.4 Constructibility implies choice
12.5 The continuum hypothesis
12.5.1 Overview
12.5.2 Mostowski's isomorphism and constructibility
12.5.3 Mostowski's isomorphism and ordinals
12.5.4 The theorem
12.6 Exercises
13. Boolean Algebras
13.1 Lattices
13.1.1 Basic properties
13.1.2 Filters, ideals and duality
13.1.3 Distributive and complemented lattices
13.2 Boolean algebras
13.3 Complete lattices and algebras
13.3.1 Complete Boolean algebras
13.3.2 Separative orderings and complete algebras
13.3.3 The completion of an ordered set
13.4 Generic filters
13.5 Exercises
14. Generic extensions of a universe
14.1 The basics
14.1.1 Boolean universes
14.1.2 The construction of a generic extension
14.2 The generic universe is a ZFC-universe
14.2.1 First axioms
14.2.2 The axiom of the power set
14.2.3 Elements and classes of a generic extension
14.2.4 The axiom of replacement for generic extensions
14.2.5 Infinity and choice
14.3 Cardinals of generic extensions
14.4 Exercises
15. Independence proofs
15.1 Pointed universes
15.2 Cohen's theorem on the CH
15.2.1 Overview
15.2.2 A counterexample to CH
15.3 Intermediate generic extensions
15.3.1 Motivation
15.3.2 Automorphisms of the algebra A
15.4 The construction of the extension
15.4.1 Another Boolean universe
15.4.2 The universe M(G,₣)
15.5 Cohen's theorem on the axiom of choice
15.5.1 Overview
15.5.2 The basic data
15.5.3 The construction of the set Z
15.5.4 A counterexample to AC
15.6 Exercises
III. Appendices
A. The NBG-Theory
A.1 Introduction
A.2 NBG-universes
A.3 NBG vs ZF
B. Logic and Set Theory
B.1 Introduction
B.2 First-order set theory
B.2.1 Language of set theory
B.2.2 Deduction
B.3 FOST and IST
B.3.1 Evaluation of formulas
B.3.2 Models of theories
B.4 The completeness theorem and consequences
B.4.1 The theorem
B.4.2 Final remarks
C. Real Numbers Revisited
C.1 Only one set of real numbers?
Biography
José Luis García is Emeritus Professor at the University of Murcia, Spain. He began his study of Mathematics at the University of Granada (Spain). After several years teaching Mathematics in secondary school, he was enrolled by the University of Murcia (Spain) as a member of the group working on algebra, led by Professor Gomez Pardo. He received his doctorate and a permanent position at the same university. He developed his research on Ring and Module Theory. He became full professor in 1991, and then served for thirty years.
