Starred proofs and appendicesExercisesI : 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 forthe 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 accountsthat are written at a level accessible to a broad range of readers, and this one is no exception. As such, it stands in stark contrast to Halmos, and in this course it will serve as a workbook to complement Halmos.The following book has also been used for this course in the past and might provide some useful additional background. It is written at a higher level than Halmos but it is also contains very substantially more detailed information. D. Goldrei, Classic Set Theory: A guided independent study. Chapman and Hall, London, 1996. ISBN 0–412–60610–0. Still further references (e.g., the text for Mathematics 11 by K. Rosen) will be given later.These course notes are designed as a further source of official information, generally ata level somewhere between the two required texts. Comments on both Halmos and Lipschutz will be inserted into these notes as they seem necessary. I.1 : Overview of the course(Halmos, Preface; Lipschutz, Preface)Set theory has become the standard framework for expressing most mathematical statements and facts in a formal manner. Some aspects of set theory now appear at nearly every level of mathematical instruction, and words like union and intersection 1have become almost as standard in mathematics as addition, multiplication, negative and zero. The purpose of this course is to cover those portions of set theory that are used and needed at the advanced undergraduate level.In the preface to Naive Set Theory, P. R. Halmos (1916 – ) proposes the following characterization of the set – theoretic material that is needed for specialized undergraduate courses in mathematics: Every mathematician agrees that every mathematician must know some set theory; the disagreement begins in trying to decide how muchis some. The purpose … is to tell the beginning student the basic set-theoretic facts … with the minimum of philosophical discourse and logical formalism. The point of view throughout is that … the concepts and methods … are merely some of the standard mathematical tools.Following Halmos, whose choice of a book title was strongly influenced by earlier writings of H. Weyl (1885 – 1955), mathematicians generally distinguish between the “naïve” approach to set theory which provides enough background to do a great deal of mathematics and the axiomatic approach which is carefully formulated in order to address tough questions about the logical soundness of the subject. We shall discuss some key points in the axiomatic approach to set theory, but generally the emphasis will be on the naïve approach. The following quotation from Halmos provides some basic guidelines:axiomatic set theory from the naïve point of view … axiomatic in that some axioms for set theory are stated and used as the basis for all subsequent proofs … naïve in that the language and notation are those of ordinary informal (but formalizable) mathematics. A more important way in which the naïve point of view predominates is that set theory is regarded as a body of facts, of which the axioms are a brief and convenient summaryThe Halmos approach to teaching set theory has been influential and has proven itself ina half century of use, but there is one point in the preface to Naive Set Theory that requires comment: In the orthodox axiomatic view [of set theory] the logical relations among various axioms are the central objects of study.An entirely different perspective on axiomatic set theory is presented in the following online site:http://plato.stanford.edu/entries/set-theoryMuch of the research in axiomatic set theory that is described in the online site involves (1) the uses of set theory in other areas of mathematics, and (2) testing the limits to which our current understanding of mathematics can be safely pushed.There is some overlap between the contents of this course and the lower level course Mathematics 11: Discrete Mathematics. Both courses cover basic concepts and terms from set theory, but there is more emphasis in the former on counting problems 2and more emphasis here on abstract constructions and properties of the real number system. A related difference is that there is more emphasis on finite sets in Mathematics11. At various points in the course it might be worthwhile to compare the treatment of topics in this course and its references with the presentation in the corresponding text forMathematics 11:K. H. Rosen, Discrete Mathematics and Its Applications (Fifth Ed.). McGraw – Hill, New York, 2003. ISBN: 0– 07293033– 0. Companion Web site: http://www.mhhe.com/math/advmath/rosen/Some supplementary exercises from this course will be taken from Rosen, and supplementary references to it will also be given in these notes as appropriate.One basic goal of an introduction to the foundations of mathematics is to explain how mathematical ideas are expressed in writing. Therefore a secondary aim of these notes (and the course) is to provide an overview of modern mathematical notation. In particular, we shall attempt to include some major variants of standard notation that are currently in use.At some points of these notes there will be discussions involving other areas of the mathematical sciences, mainly from lower level undergraduate courses like calculus (for functions of one or several variables), discrete mathematics, elementary