Set Theory: An Introduction to Axiomatic Reasoning

Description

Contemporary students of mathematics differ considerably from those of half a century ago. In spite of this, many textbooks written decades ago, and now considered to be “classics”, are still prescribed for students today. These texts are not suitable for today’s students. This text is meant for and written to today’s mathematics students. Set theory is a pure mathematics endeavor in the sense that it seems to have no immediate applications; yet the knowledge and skills developed in such a course can easily branch out to various fields of both pure mathematics and applied mathematics. Rather than transforming the reader into a practicing mathematician, this book is more designed to initiate the reader to what may be called “mathematical thinking” while developing knowledge about foundations of modern mathematics. Without this insight, becoming a practicing mathematician is much more daunting. The main objective is twofold. The students will develop some fundamental understanding of the foundations of mathematics and elements of set theory, in general. In the process, the student will develop skills in proving simple mathematical statements with “mathematical rigor”. Carefully presented detailed proofs and rigorous chains of logical arguments will guide the students from the fundamental ZFC-axioms and definitions to show why a basic mathematical statement must hold true. The student will recognize the role played by each fundamental axiom in development of modern mathematics. The student will learn to distinguish between a correct mathematical proof and an erroneous one. The subject matter is presented while bypassing the complexities encountered when using formal logic. 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