Introduction to Modern Set Theory

1 Introduction to Modern Set TheoryJudith RoitmanDecember 6, 20111for my father, who loved mathematics1 Revised Edition Copyright 2011 by Judith Roitman, This book is licensed for use under a Creative CommonsLicense(CC BY-NC-ND ). You may download, share, and use this work at no charge, but may not modify norsell of ContentsPreface31. Partially ordered The structure of partially ordered Equivalence Well-ordered Mathematical Filters and Exercises2. Theories and their First-order Models and Exercises3. The axioms, part Why axioms? The language, some finite operations, and the axiom of Models of Models of Cartesian Union, intersection, Models of union and last3,6,1 Models of Power Models of power Models of Exercises24.

2 Regularity and Regularity, part I4,2 Transitive A first look at Regularity, part Equivalents to Models of regularity and Embedding mathematics into set Exercises5. Infinite Cardinality with Ordinal Cardinal Infinite operations and more Exercises6. Two models of set A set model for The constructible Exercises7. Semi-advanced set Partition Measurable Cardinal invariants of the CH and Stationary sets and Exercises4 PrefaceWhen, in early adolescence, I first saw the proof that the real numbers were uncountable, I washooked. I didn t quite know on what, but I treasured that proof, would run it over in my mind,and was amazed that the rest of the world didn t share my enthusiasm.

3 Much later, learning thatset theorists could actually prove some basic mathematical questions to be unanswerable, and thatlarge infinite numbers could effect the structure of the reals the number line familiar to all of usfrom the early grades I was even more astonished that the world did not beat a path to the settheorist s years later than I care to admit (and, for this revision, yet another twenty years later),this book is my response. I wrote it in the firm belief that set Theory is good not just for settheorists, but for many mathematicians, and that the earlier a student sees the particular point ofview that we call Modern set Theory , the is designed for a one-semester course in set Theory at the advanced undergraduate or beginninggraduate level.

4 It assumes no knowledge of logic, and no knowledge of set Theory beyond the vaguefamiliarity with curly brackets, union and intersection usually expected of an advanced mathematicsstudent. It grew out of my experience teaching this material in a first-year graduate course atthe University of Kansas over many years. It is aimed at two audiences students who areinterested in studying set Theory for its own sake, and students in other areas who may be curiousabout applications of set Theory to their field. While a one-semester course with no logic as aprerequisite cannot begin to tell either group of students all they need to know, it can hope to laythe foundations for further study.

5 In particular, I am concerned with developing the intuitions thatlie behind Modern , as well as classical, set Theory , and with connecting set Theory with the rest , three features are the full integration into the text of the study of models of set Theory ,the use of illustrative examples both in the text and and in the exercises, and the integration ofconsistency results and large cardinals into the text when appropriate, even early on (for example,when cardinal exponentiation is introduced). An attempt is made to give some sense of the historyof the subject, both as motivation, and because it is interesting in its own first chapter is an Introduction to partial orders and to well-ordered sets, with a nod toinduction onN, filters, and ideals.

6 The second chapter is about first-order theories and their models;this discussion is greatly extended from the first edition. Without becoming too formal, this chaptercarefully examines a number of theories and their models, including the Theory of partially orderedsets, in order to provide a background for discussion of models of the various axioms of set third chapter introduces all of the axioms except regularity and choice, formally definesthe natural numbers, and gives examples of models of the axioms, with an emphasis on standardmodels (in which the symbol is interpreted by the real relation ). The fourth chapter discussestransitive sets, regularity, ordinals, and choice and, in its last section, gives a taste of how to embedstandard mathematics within set and ordinal numbers are the subject of chapter five.

7 Chapter six discussesV,V where is inaccessible, andL. Chapter seven introduces infinite combinatorics: partition calculus, trees,measurable cardinals, CH. Martin s axiom, stationary sets and , and cardinal invariants of brief word about formality. The first chapter is written in ordinary mathematical style5without set-theoretical formality (compare the definition of partial order in section with theformal definition in section ). The reader is assumed to be familiar with set-theoretic notation asfound in most advanced mathematical texts, and we will make use of it throughout. The reader isalso assumed to be familiar with the standard body of basic mathematics, , the basic propertiesof the natural numbers, the integers, the rationals, and the PreliminariesThe reader is probably used to picking up a mathematical textbook and seeing a first chapterwhich includes a section with an approximate title of Set-theoretical prerequisites.

8 Such a chapterusually contains a quick review or an overview of the relevant set Theory , from something as simpleas the definition of the union of two sets to something as complicated as the definitions of countableand uncountable sets. Since this is a set Theory text, we reverse the usual procedure by puttingin the first chapter some mathematics that will prove essential to the serious study of set Theory :partially ordered and linearly ordered sets, equivalence relations, well-ordered sets, induction andrecursion, filters and Why these topics?The spine of the set-theoretic universe, and the most essential class of objects in the study of settheory, is the class of ordinals.

9 One of the basic properties of an ordinal is that it is a well-orderedset. An acquaintance with various examples and properties of well-ordered sets is essential to thestudy of of the basic techniques of set Theory are transfinite induction and transfinite recursion,which are grounded in induction and recursion on the natural set Theory is applied to the rest of mathematics, the methodology often used is to reducethe original question to a question in the area known as infinite combinatorics. The combinatoricsof partially ordered sets (especially those known as trees, see chapter 7) are particularly important,and partially ordered sets are crucial to the major technique of proving consistency , filters and ideals are not only important combinatorial objects, but are essential to thetheory of large the choice of topics in this Partially ordered setsPartially ordered sets underlie much of the combinatorics we will is a partial order on a setXiff for allx,y,z XP1(Reflexive axiom)x (Antisymmetric axiom) Ifx yandy xthenx= (Transitive axiom) Ifx yandy zthenx shorthand, we sayx < y(xis strictly less thany) ifx yandx6=y.

10 If partiallly ordersX, we callXa partially ordered set under and say<strictly ordersX. As will become clearfrom the examples, a set can have many different partial orders imposed upon are some the set of natural numbersN={0,1,2, }. Forn,k Nwe definen this a partial order?Check for P1: Everyndividesn, so eachn for P2: Ifndivideskthenn k(where is the usual order). Ifkdividesn, thenk n. We know that ifn kandk nthenn=k. Hence ifndivideskandkdividesn, thenn= for P3:n Dkiffk=infor Dmifm=jkfor somej. So ifn Dk Dmthere arei,jwithm=jk=jin. Hencen that thek,jwhose existence was needed for the proof of P3 must come fromN. Thefact that 2 =23(3) does not imply that 2 D3.