Axiomatic Set Theory

Rating:
Summary: EXERCISES
Review: I skim this book one day while looking for some reference books at my local bookstore.The clarity of the book makes it a good book for complementing set theory courses.The examples are given in a consice manner without obstructing the learning material to the readers. However, the lack of answers to the given problems makes the book unfitting to first time readers who may want to learn the subject just for plain curiosity; none of the steps are hinted for solving any problem. It is unfair having to buy this book because readers buying this book are expected to be experts in the field. Rather, it should also consider general science readers who have interest for the subject and want to learn the material. I think having full answers to problems allows all readers to have a good understanding of the subject at hand since it clarifies the bridge of ideas that mathematicians are trying to let the world see. Furthermore, it motivates readers to critically think challenging problems once enough practice has been establish through the ones with answers. However, because there is a lack of communication and guidance to achieve a full census with the subject, this book has a mild sour taste. So in hopes that the author of this book improves the book for the benefit of his readers, "Please provide answers next time."

Rating:
Summary: Still interesting...and still important.
Review: One does not hear about set theory too much these days, no doubt due to the de-emphasis of foundational discussions in mathematics. Foundational questions of course were the focus of much attention in mathematics in the early twentieth century, this taking place because of the many paradoxes in set theory and due to the influence of the philosophers. Set theory, the theory of types, and mathematical logic are still very important though in computer science and in artificial intelligence, due to the needs in these fields for knowledge representation, computational models of intelligence, and automated reasoning. This book could serve to introduce these topics or as an historical reference to the issues as they were hotly debated in the last century.

The first chapter gives an informal introduction to the notion of a set, first-order predicate logic (notions of bound and free variables and quantification), and the Zermelo-Fraenkel axioms of set theory. The author describes the difficulties in the "axiom of abstraction" in the writings of Frege as pointed out by Bertrand Russell. It is pointed out that the axiom of abstraction is in fact an infinite collection of axioms, thus motivating the concept of an "axiom schema". The axiom schema that is used explicitly in the book is the "axiom schema of separation" due to Ernst Zermelo, which he formulated in order to make precise the notion of a statement as being "definite". More of the set-theoretic paradoxes are discussed, along with their classification due to F.P. Ramsey into "linguistic" and "semantical" ones.

The advantage of an older book on set theory is that more of the underlying details are explained, instead of just being formally developed. The author gives a thorough discussion of the concepts throughout the book, beginning with an organized development in chapter 2. He begins immediately with discussing the distinction between the object language and metalanguage, and the symbols to be used in the object language: constants, variables, logical connectives, quantifiers, and grouping symbols. These symbols are used to construct formulas, a subclass of which, the primitive formulas, are defined recursively, and which all formulas in the object language can be expressed in terms of. Throughout the book though the author uses additional notation that allows formulas not to be written in terms of primitive formulas. This is done to make the text more readable, but he requires that the added notion satisfy the criterion of eliminability and non-creativity. The notion of a set is defined formally, and then the axiom of extensionality, which gives a criterion for two sets being equal, and the axiom schema schema of separation. The pairing axiom, which gives the existence of a non-empty set; the sum axiom, which gives the existence of the union of a family of sets; the power set axiom, which gives the notion of the set of all subsets of a set; and the axiom of regularity, which prohibits infinite descending sequences of sets, are all discussed in detail.

Chapter 3 treats relations and functions, so important not only in mathematics but in computer science, especially in the theory of relational databases. Then in chapter 4, the author begins a study of cardinality and the cardinal numbers, proving that the finite cardinal numbers have the properties of the natural numbers, as one would expect. The author is careful to point out the need for the axiom of cardinal numbers in this study. Chapter 5 then goes into the theory of ordinal numbers, wherein it is emphasized that no special axioms are needed for the development of this theory. The author is also careful to note the special problems that arise in defining the arithmetic of natural numbers, such as defining addition recursively without using set theory. But including the apparatus of set theory does allow the replacement of the recursive definition by a proper definition. The axiom of infinity is brought in to permit the construction of arithmetical operations as certain sets. The theory of denumerable sets is then discussed, followed by one of the most fascinating concepts in all of mathematics: the theory of transfinite and infinite cardinals.

The author then shows that set theory can allow the construction of the real numbers, which takes place after the construction of the rational numbers. The famous "Dedekind cut" is discussed, along with the method of Cantor, which defines real numbers as equivalence classes of Cauchy sequences of rational numbers. The author uses the Cantor approach in the rest of the book. He also proves the famous Cantor theorem on the non-denumerability of the real numbers, and gives a brief discussion of the Continuum Hypothesis.

Chapter 8 then gives an overview of the fascinating topics of transfinite induction and ordinal number theory. Recursion theory makes its appearance again in the transfinite recursion for ordinal numbers, using the axiom schema of replacement. The non-commutativity of ordinal addition and multiplication is brought out, and the falsity of Fermat's Last Theorem and Goldbach's Hypothesis in ordinal number theory is shown. The author then shows to what extent cardinal number theory can be done without using special axioms by defining cardinal numbers as initial ordinals. The axiom of choice however is needed to show that every set has a cardinal number. The author then restates the Zermelo-Fraenkel axioms in their final form at the end of the chapter.

The final chapter gives an overview of the most controversial topic in all of set theory, if not in all of mathematics: the axiom of choice. The author shows that the use of this axiom allows one to prove that an infinite set has a denumerable subset, and he shows the equivalence of the axiom of choice with the numeration theorem, the well-ordering theorem, Zorn's lemma, and the law of trichotomy. The counterintuitive Banach-Tarski paradox is discussed, and the author shows the existence of axioms which imply the axiom of choice.

Rating:
Summary: An Excellent Text for Self-Study
Review: This book presents a rigorous, axiomatic development of classic set theory, introducing the axioms as needed and founding nearly all results upon theorems derived earlier in the book (or on the axioms themselves). It is genuinely gratifying to see the development proceed in such a regimented fashion, from basic sets to natural numbers to reals, and then on to transfinite induction and the axiom of choice. There are numerous exercises; no answers are provided, but the intelligent reader who proceeds carefully should not find this a hindrance. It is however, not a modern book; readers who want to understand current ideasin set theory (inaccessible, supercompact cardinals, etc.) should look elsewhere.

Rating:
Summary: Decent intro book.
Review: This is a basic introduction to axiomatic set theory. You dont need much experience with informal set theory or formal logic to begin it. The book is rigorous and follows a definition - theorem - proof format, broken with clear exposition and historical notes. Enough formal mathematical logic is introduced only to express the axioms (that is, formal proof systems are not used or discussed). He uses the ZF (Zermelo/Fraenkel) system and gives footnote comparisons to the NBG (von Neumann/Bernays/Godel) system. In chapter 4 he introduces a special axiom (outside of ZF) to simplify the development of cardinal arithmetic, and this involves the addition of a new primitive notion. All theorems relying on this special axiom are clearly marked. While this admirably allows Suppes to avoid employing the axiom of choice or developing a much more complicated strictly ZF-based construction of the cardinals, it does make the book unacceptable for more advanced readers. In chapter 6 he gives a detailed construction of the rational numbers and the real numbers (using Cauchy sequences). He uses only the axiom of separation until the axiom of replacement becomes necessary. He does a good job explaining why each axiom is needed and how it arose historically. The book is comparable to Monk's _Introduction to Set Theory_, though a little easier and less advanced.