1st Edition

# Intuitive Axiomatic Set Theory

By José L Garciá Copyright 2024
362 Pages
by Chapman & Hall

362 Pages
by Chapman & Hall

Also available as eBook on:

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

8.5.2 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

10.2.2 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.