Offering: Foundations of Modern Mathematics

Titles: Foundations of Modern Mathematics

By Dr. Lee Keener

Preface

This text was developed for use in the courses Foundations of Modern Mathematics I and II, taught over a number of years at the University of Northern British Columbia. The audience has consisted primarily of second-year mathematics majors. Typically, these students have completed a two-semester sequence in one-variable calculus and have completed, or are concurrently enrolled in, a course in linear algebra.

In a formal sense, the text may be read with only high school mathematics as required background material. However, readers with this limited exposure to mathematics will find the material very challenging. In particular, the presentation of the fundamentals of differential and integral calculus is rigourous but extremely terse; some previous exposure to the topics is implicitly assumed. More generally, the reader is expected to have a certain level of mathematical maturity, to be comfortable reading proofs of theorems, and to be facile with mathematical notation. Even students with appropriate backgrounds may find the text difficult. At some point in a mathematician’s training it becomes necessary to wade in and get one’s hands dirty, to think for oneself, and to deal with demanding notational and conceptual issues. There is really no easy way around this educational crux.

There are a few theorems with proofs sufficiently difficult that I have usually omitted these proofs in the above-mentioned courses. These include the least upper bound property of the real numbers, the result that every complete ordered field is isomorphic to the real numbers, the higher dimensional versions of the Heine-Borel and Bolzano-Weierstrass Theorems, and a theorem on differentiating power series. Even near the start of the text there are theorems with tough (or at least notationally challenging) proofs. For example, Theorem 1.1.3 is really a result about the use of parentheses, but to give it the required generality necessitates a fairly awkward collection of definitions. I usually omit the proof. I also omit the proof to Theorem 1.2.10 on the isomorphism between finite geometries of order 3.

I have tried very hard to avoid superficial development of the topics. Every section contains at least one important and non-trivial result. On the other hand, the treatment of a given topic is rarely comprehensive.Usually I go into enough depth to obtain results that are needed later and to provide a sense of the flavour and variety of the material and of the manner in which apparently remote areas of study support each other. However, the coverage of metric topology is more complete as befits its central role in many branches of mathematics.

The text is divided into two parts or chapters. A version of the text that includes only selected sections, from Chapter 1, is also available. The first chapter includes all of the core material and focuses primarily on the development of the five key number systems: the natural numbers, the integers, the rationals, the reals, and the complexes. The culminating theorem is the Fundamental Theorem of Algebra. Along the way, a number of topics are explored tangentially. For example, the above-mentioned section on metric topology develops some needed results for the proof of the Fundamental Theorem. But a lot more is developed as well. The second chapter of the text comprises a selection of optional topics. In two semesters I cover all of Chapter I and use Chapter II as a basis for student projects and presentations.

Some theorems, such as the Fundamental Theorem of Algebra, can be proved with powerful methods that are not considered in this book. I have not hesitated to provide an elementary proof in place of a more efficient (but inaccessible) alternative. Note that “elementary” is not a synonym for “easy”! Most proofs and developments are done in a manner that is fairly standard in the literature but there are exceptions.

In naming theorems, concepts, etc., and in choosing notation, I have tried to make the decision consistent with the most common usage. A possible deviation from this pattern is in the definition of countable which I insist implies infinite. The majority of texts consider finite sets to be countable, though Rudin [40] is a notable exception. I am convinced that the day-to-day usage of working mathematicians is more usually in line with the definition given in this text. By contrast, I have not always provided the “standard” proofs of well-known theorems. For example, the proofs of the Bolzano-Weierstrass and Heine-Borel Theorems are not the usual ones in the literature and the approach to the Fourier Theorem is perhaps new. Whether there is any advantage to these alternatives the reader must judge. In some cases the usual proofs are treated in the exercises. A few definitions are local in the sense that they serve a temporary purpose in a development and are not used outside the context of this book. Local definitions are flagged with the notation “(l)” in the index.

The exercises, of which there are more than 200, are very frequently based on material that has not been fully developed in the text exposition proper. A few are quite difficult or open-ended. These are starred.

The text is self-contained in the following sense: Certain results from set theory and logic are assumed. A brief survey of the set theory essential for reading the book is provided in the first section. A second companion volume is planned which will address formal systems, mathematical logic, axiomatic set theory, and model theory. Since this book is not yet available, a short appendix on axiomatic set theory is provided to help fill the gap. Except for this, for the development of a series representation of π, and for some looseness in the treatment of planar graphs in Section 2.6, every result that is used in Sections 1.1-2.9 is proved in the text. Section 2.10 on Euclidean and non-Euclidean geometries concentrates on issues of independence and consistency. A linear development of the corpus of theorems of Euclidean and neutral geometry is impractical in the context of this text and is omitted. Similarly, Section 2.11 on non-standard analysis is only a sketch of a rich and important field. In a good number of places certain details of a proof are relegated to the exercises. And there are a few places where a theorem, such as the Prime Number Theorem, is stated but not proved. But these results are not used subsequently.

The sections of Chapter I are meant to be read in the order given and to a great extent each of these sections relies on the previous sections. Exceptionally, the section on finite geometries, though referred to in Section 1.11, is not essential for understanding any other section. It is provided for practice in constructing proofs and in thinking abstractly.

The sections of Chapter II are based on Chapter I but are largely independent of each other. The only exceptions are that Section 2.2 depends to some degree on Section 2.1, Section 2.5 requires, in the proof of the Fourier Theorem, the trigonometric form of the Weierstrass Approximation Theorem from Section 2.4, Section 2.7 requires the terminology and one simple result from Section 2.6, and Section 2.8 uses some terminology from Section 2.2.

There is a great deal of variance in the length and difficulty of the sections of Chapter II. Section 2.8 on measure theory is particularly hard, perhaps the hardest section of the text.

I have relied on a number of authors in structuring the approaches to the topics. Herstein’s Topics in Algebra [20] was an important guide for much of the algebraic material. Rudin’s marvellous Principles of Mathematical Analysis [40] was of similar value for the analysis material, except for the measure theory, where I followed in large part the general development in Royden’s Real Analysis [38]. In a few places, the debt is greater and the approach followed a bit more slavish. I mention Niven and Zuckerman’s An Introduction to the Theory of Numbers [32] for the first part of Section 1.4, Greenberg’s Euclidean and Non-Euclidean Geometries [14] for Section 2.9, Wilder’s Introduction to the Foundations of Mathematics [53] for some of Section 1.2, Bressoud and Wagon’s A Course in Computational Number Theory [8] for the exercises on Chebyshev’s Theorem in Section 2.3, and Halmos’s Naive Set Theory [18] for much of Section 2.12. The brief summary in Section 2.11 was suggested by the (even briefer) summary in Freshman Calculus by Bonic et al [5]. The proof of the Schr ̈ der-Bernstein Theorem in Section 2.12 was adapted from Goldrei [13]. The book of Chartrand and Lesniak [9] was useful in organizing Section 2.6.

The choice of topics is of course somewhat idiosyncratic. There may be a bit of bias towards analysis, which is my field. In fact there is enough material in the text to comprise a semester course in real analysis.

Exactly what should be part of a course in foundations of modern mathematics could be hotly debated. Theterms might suggest an emphasis on set theory and logic and indeed this is the focus of the planned Volume II. I believe that it is best to put off this material until students have a better grasp of the relatively more concrete structures that can be built on the results and ideas to be developed later and more fully in this second volume.

A word about style may be in order. When a concept is officially defined (either in a displayed definition or theorem, or in the course of the narrative) it is given in boldface. A few of the more important definitions appear more than once, often in somewhat different contexts. Theorems or Lemmas with traditional names have those names shown in boldface. Earlier references to a concept not yet defined are shown in italics. Italics are also used for emphasis. Symbols for the various number systems are shown in boldface as for example in R, which represents the set of real numbers. Vectors in Euclidean space are also shown in boldface and boldface is used, in Section 2.11 only, to indicate non-standard real numbers. Except for headings, these are the only uses of boldface. A reference to an exercise depends on the section number. For example, Problem 2 of Exercises 1.4.3 refers to Problem 2 in Section 1.4.3. For those unfamiliar with the Greek alphabet, it is presented, with a few comments, in Appendix III (Section 2.14).