1st Edition

Intuitive Axiomatic Set Theory

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

    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?








    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.