MATHEMATICS 144 SET THEORY FALL 2012 VERSION Table of Contents I. General considerations.……………………………………………………………………………………………………….1 1. Overview of the course…………………………………………………………………………………………………1 2. Historical background and motivation………………………………………………………….………………4 3. Selected problems………………………………………………………………………………………………………13 I I. Basic concepts. ………………………………………………………………………………………………………………….15 1. Topics from logic…………………………………………………………………………………………………………16 2. Notation and first steps………………………………………………………………………………………………26 3. Simple examples…………………………………………………………………………………………………………30 I I I. Constructions in set theory.………………………………………………………………………………..……….34 1. Boolean algebra operations.……………………………………………………………………………………….34 2. Ordered pairs and Cartesian products……………………………………………………………………… ….40 3. Larger constructions………………………………………………………………………………………………..….42 4. A convenient assumption………………………………………………………………………………………… ….45 I V. Relations and functions ……………………………………………………………………………………………….49 1.Binary relations………………………………………………………………………………………………………… ….49 2. Partial and linear orderings……………………………..………………………………………………… ………… 56 3. Functions…………………………………………………………………………………………………………… ….…….. 61 4. Composite and inverse function.…………………………………………………………………………… …….. 70 5. Constructions involving functions ………………………………………………………………………… ……… 77 6. Order types……………………………………………………………………………………………………… …………… 80 i V. Number systems and set theory …………………………………………………………………………………. 84 1. The Natural Numbers and Integers…………………………………………………………………………….83 2. Finite induction and recursion………………………………………………………………………………… …89 3. Finite sets…………………………………………………………………………………………………………………..95 4. The real number system……………………………………………………………………………………………100 5. Further properties of the real numbers…………………………………………………… …… …………..104 APPENDIX. Proofs of results on number expansions…………………………………………….. …… 113 V I. Infinite constructions in set theory ………………………………………………..………………… …. .….121 1. Operations on indexed families……………………………………………………… ……… ……… ……………. 121 2. Infinite Cartesian products………………………………………………………………………… …… …………… 123 3. Transfinite cardinal numbers………………………………………………………………………… ……… ……129 4. Countable and uncountable sets…………………………………………………………………………..……132 5. The impact of set theory on mathematics………………………………………………………… …….…… 146 6. Transfinite recursion…………………………………..…………………………………………………… …..……148 V I I. The Axiom of Choice and related properties ………………………………………………………..158 1. Some questions………………………………………………………………………………………..…………….. …159 2. Extending partial orderings……………………………………………………………………………………… …162 3. Equivalence proofs……………………………………………….………………………………………………. ……166 4. Additional properties……………………………………………………………..…………………………… .. ……168 5. Logical consistency……………………………………………….……………………………………………… …….172 6. The Continuum Hypothesis……………………………………………………………………….……… …..……177 V I I I. Set theory as a foundation for mathematics ………………………………………………..……... 180 1. Formal development of set theory………………………………………………..………………….……..…180 2. Simplifying axioms for number systems……………………………………………..……… …….……...….182 3. Uniqueness of number systems……………………………………………..…………… .… ………….. ….. …..192 4. Set theory and classical geometry………………………………………………………….………….…..……203 (iv + 210 pages) NOTE. This document is meant for instructional purposes involving students and instructors at the University of California at Riverside and is not intended for public distribution . Please respect these intentions when downloading it or printing it out . ii PREFACE This is a slightly modified set of notes from the most recent time I taught Mathematics 144 , which was during the Fall 2006 Quarter . There are only a few minor revisions and insertions, with updated biographical information and links as needed . Since clickable Internet references appear frequently in the notes , I have also included my standard policy remarks about the use of such material. The official main text for this course was the book on set theory in the Schaum ’s Outline Series by S. Lipschutz , but for several abstract or technical issues there are references to previously used course texts by P. Halmos and D. Goldrei (see page 1 for detailed information on all three of these books). The online directory for the 2006 course http://math.ucr.edu/~res/math144/ also contains several files of exercises and solutions based upon the notes . Most of the set – theoretic notation is extremely standard , and we shall also employ some frequently used conventions for using “blackboard bold letters ” and other characters to denote familiar sets and number systems : Ø empty set NNN natural numbers = nonnegative integers ZZZ (signed) integers QQQ rational numbers RRR real numbers CCC complex numbers n Similarly , we shall use RRR to denote the usual analytic representation of Euclidean or Cartesian n – [dimensional ] space in terms of coordinates (x1, … , xn), where the xi ’s are all real numbers . As in calculus , if a and b are real numbers or ± ∞∞∞ with a < b, we define intervals as follows : |Notation | Type |of |interval Defining |inequalties . (a, |b) open a < x < b | . [a, |b] closed a ≤ x ≤ b . (a,|b] half open a < x ≤ b . [a,|b) half open a ≤ x < b . Reinhard Schultz Department of Mathematics University of California , Riverside December , 2012 iii Comments on Internet resources Traditional printed publications in mathematics are normally filtered through an editorial reviewing process which checks their accuracy (not perfectly , but for the most part very reliably) . Some widely used Internet sources maintain similar standards (for example , most of the sites supported by recognized academic institutions) , but others have far more lenient standards , and this fact must be acknowledged . Probably the most important single example is the Wikipedia site : http://en.wikipedia.org/wiki/Main_Page The Wikipedia site contains an incredibly large number of articles , with extensive information on a correspondingly vast array of subjects . The articles are written by volunteers , and in most cases they can be edited by anyone with access to the Internet , including some individuals whose views or understanding of a subject may be highly controversial or simply unreliable . This issue has been noted explicitly by Wikipedia in its articles on itself , and in particular the following discuss the matter in some detail . http://en.wikipedia.org/wiki/Wikipedia http://en.wikipedia.org/wiki/Reliability_of_Wikipedia Since a few documents in this directory make references to Wikipedia articles , the underlying policies and reasons for doing so deserve to be discussed . First of all , despite the justifiable controversy surrounding the reliability of some online Wikipedia articles , the entries for standard , well – established topics in the sciences are generally very reliable , and the ones cited in the course notes were specifically checked for accuracy before they were cited . As such , they are inserted into these notes as convenient but reliable online alternatives to more traditional library references strictly on a case by case basis . Consequently , this usage should not be interpreted as a blanket policy of acceptance for all such articles , even in the “hard ” sciences . In general , it is best to think of Wikipedia articles as merely first steps in gathering information about a subject and not as substitutes or replacements for more authoritative (printed or electronic) references in term papers or scholarly articles . All statements in Wikipedia articles definitely should be checked independently using more authoritative sources . In any discussion of Internet references , some comments about World Wide Web searches using Google (or other search engines) are also appropriate . The extreme popularity and wide use of Google searches clearly show their value for all sorts of purposes . Of course , it is important to remember that search engines are designed to make money and that profit motives might affect the results of searches , but usually this is not a problem for topics in the sciences . Most of the time search engines are very reliable at listing the best references first , but this is not always the case , and therefore it is strongly recommended that a user should normally go beyond the first page of 10 search results . As a rule , it is preferable to look at the top 20, 50 or even 100 results . iv I : General considerations This is an upper level undergraduate course in set theory . There are two official texts . P. R. Halmos , Naive Set Theory (Undergraduate Texts in Mathematics) . Springer – Verlag , New York , 1974 . ISBN : 0–387 –90092 –6. This extremely influential textbook was first published in 1960 and popularized the name for the “working knowledge ” approach to set theory that most mathematicians and others have used for decades . Its contents have not been revised , but they remain almost as timely now as they were nearly fifty years ago . The exposition is simple and direct . In some instances this may make the material difficult to grasp when it is read for the first time , but the brevity of the text should ultimately allow a reader to focus on the main points and not to get distracted by potentially confusing side issues . S. Lipschutz , Schaum's Outline of Set Theory and Related Topics (Second Ed.) . McGraw –Hill , New York , 1998 . ISBN : 0–07 –038159 –3. The volumes in Schaum ’s Outline Series are designed to be extremely detailed accounts that are written at a level accessible to a broad range of readers , and this