Axioms and Set Theory A first course in Set Theory Robert Andr´e Robert Andr´e c 2014 ISBN 978-0-9938485-0-6 Revised 21/07/06 A` Jinxia, Camille et Isabelle “Everything has beauty, but not everyone sees it.” Confucius i Preface A set theory textbook can cover a vast amount of material depending on the mathematical background of the readers it was designed for. Selecting the material for presentation in this book often came down to deciding how much detail should be provided when explaining concepts and what constitutes a reasonable logical gap which can be independently filled in by the reader. The initial chapters of this book will appeal to students who have little expe- rience in proving mathematical statements, while the last chapters, significantly denser in subject matter, will appeal considerably more to senior undergraduate or graduate students. Choice of topics and calibration of the level of communication is based on the estimated mathematical fluency of the target students. The first part of this book (Chapters 1 to 21) is written for intermediate level math major students in mind. At this level, most students have not yet been exposed to the mathematical rigor normally found in most textbooks in set theory. The pace at which new concepts are introduced at the beginning is what some may subjectively consider as being quite “leisurely”. The meaning of mathematical statements is explained at length and their proofs presented in great detail. As the student progresses through the course, he or she will develop a better understanding of what con- stitutes a correct mathematical proof. To help attain this objective, numerous examples of simple straightforward proofs are presented as models throughout the text. The subject material is subdivided into ten major parts. The first few are themselves subdivided into “bite-size” chapters. Smaller sections allow students to test their under- standing on fewer notions at a time. This will allow the instructor to better diagnose the understanding of those specific points which challenge the students the most, thus helping to eliminate obstacles which may slow down their progress later on. Each chapter is followed by a list of Concepts review type questions. These questions highlight for students the main ideas presented in that section and help them deepen un- derstanding of these concepts before attempting the exercises. The answers to all Concept review questions are in the main body of the text. Attempting to answer these questions will help the student discover essential notions which are often overlooked when first exposed to these ideas. Textbook examples will serve as solution models to most of the exercise questions at the end of each section. Exercise questions are divided into three groups: A, B and C. The answers to the group A questions normally follow immediately from definitions and theorem statements presented in the text. The group B questions require a deeper understanding of the concepts, while the group C questions allow the students to deduce by themselves a few consequences of theorem statements presented in the text. The course begins with an informal discussion of primitive concepts and a presentation of the ZFC axioms. We then discuss, in this order, operations on classes and sets, relations on classes and sets, functions, construction of numbers (beginning with the natural numbers followed by the rational numbers and real numbers), infinite sets, cardinal numbers and, finally, ordinal numbers. It is hoped that the reader will eventually perceive the ordinal ii numbers as a natural logical extension of the natural numbers and as the “spine of set theory” like many authors described them. Towards the end of the book we present a − brief discussion of a few more advanced topics such as the Well-ordering theorem, Zorn’s lemma (both proven to be equivalent forms of the Axiom of choice) as well as Martin’s axiom. Finally, we briefly discuss the Axiom of regularity and a few of its implication. A brief and very basic presentation of ordinal arithmetic properties is then given. The pace and level of abstraction increase considerably when ordinals are introduced. It is hoped that everything which is presented before this point will allow the students to mas- ter various proof techniques while simultaneously developing a feeling for what constitutes the essence of set theory. The format used in the book allows for some flexibility in how subject matter is presented, depending on the mathematical maturity of the audience or the pace at which the students can absorb new material. A determining factor may be the amount of practice that students require to understand and produce correct mathematical proofs. Some instructors may decide to use the first twenty chapters of the book as a text for an “Introduction to mathematical proofs” course. Students who already possess a substantial amount of mathematical background may feel they can comfortably skip many chapters without loss of continuity, since these contain notions which are well-known to them. The following order sequence will allow readers with the required background to advance more quickly to the meat of the textbook: Chapter 1 on the topic of the ZFC-axioms can be immediately followed by chapters 13 and 14 on the topic of natural numbers, chapters 18 to 22 on the topic of infinite sets and cardinal numbers followed by chapters 26 to 29, 32 and 33, on ordinals, and finally, chapters 30 and 31 on the axiom of choice and the axiom of regularity. As we all know, any textbook, when initially published, will contain some errors, some typographical, others in spelling or in formatting and, what is even more worrisome, some mathematical. Many readers of the text are required to help weed out the most glaring mistakes. If you happen to be a reader who has carefully studied a chapter or two of the book please feel free to communicate to me, by email, any errors you may have spotted, with your name and chapters reviewed. In the preface of further versions of the book, I will gladly acknowledge your help. This will be much appreciated by this writer as well as by future readers. It is always more pleasurable to study a book which is error-free. Robert Andr´e University of Waterloo, Ontario [email protected] ISBN 978-0-9938485-0-6 Contents I Axioms and classes 1 Classes,setsandaxioms ............................. 1 2 Constructingclassesandsets . .. 13 II Class operations 23 3 Operationsonclassesandsets. .. 25 4 Cartesianproducts................................ 34 III Relations 43 5 Relationsonaclassorset ............................ 45 6 Equivalencerelationsandorderrelations. ....... 51 7 Partitionsinduced byequivalence relations . ....... 62 8 Equivalenceclassesandquotientsets . ..... 68 IV Functions 75 9 Functions: Aset-theoreticdefinition . ..... 77 10 Operationsonfunctions . ... ... ... ... ... .. ... ... ... 85 11 Imagesandpreimagesofsets . 93 12 Equivalence relations induced by functions . ........ 98 V From sets to numbers 105 13 Thenaturalnumbers............................... 107 14 Thenaturalnumbersasawell-orderedset. ..... 124 15 Arithmeticofthenaturalnumbers . ... 134 16 The integers Z and the rationals Q ....................... 144 17 Dedekindcuts: “Realnumbersareus!”. .... 154 v VI Infinite sets 165 18 Infinitesetsversusfinitesets. .... 167 19 Countableanduncountablesets. ... 179 20 Equipotenceasanequivalencerelation. ...... 189 21 TheSchr¨oder-Bernsteintheorem. ..... 204 VII Cardinal numbers 211 22 Anintroductiontocardinalnumbers.. ..... 213 23 Addition and multiplication in C ......................... 222 24 Exponentiationofcardinalnumbers. ..... 229 25 On sets of cardinality c .............................. 238 VIII Ordinal numbers 249 26 Well-orderedsets................................. 251 27 Ordinal numbers: Definitionand properties. ....... 266 28 Propertiesoftheclassofordinalnumbers. ....... 283 29 Initialordinals: “Cardinalnumbers are us!” . ........ 302 IX More on axioms: Choice, regularity and Martin’s axiom 327 30 Axiomofchoice.................................. 329 31 Regularityandthecumulativehierarchy . ...... 340 32 Martin’saxiom .................................. 361 X Ordinal arithmetic 369 33 Ordinalarithmetic:Addition. .... 371 34 Ordinal arithmetic: Multiplication and Exponentiation. ........... 379 Appendix A: Boolean algebras and Martin’s axiom. ....... 391 Appendix B: List of definitions and statements.. ........ 405 Bibliography....................................... 440 Index........................................... 441 Part I Axioms and classes Part I: Axioms and classes 1 1 / Classes, sets and axioms. Summary. In this section we discuss axiomatic systems in mathematics. We explain the notions of “primitive concepts” and “axioms”. We declare as prim- itive concepts of set theory the words “class”, “set” and “belong to”. These will be the only primitive concepts in our system. We then present and briefly dis- cuss the fundamental Zermelo-Fraenkel axioms of set theory. 1.1 Contradictory statements. When expressed in a mathematical context, the word “statement” is viewed in a specific way. A mathematical statement is a declaration which can be characterized as being either true or false. By this we mean that if a statement is not false, then it must be true, and vice-versa. There exists a predetermined set of rigorous logical rules which can be used to help determine the true or false value of such statements. Whether one does mathematics as an expert or as a beginner, these elementary rules of logic