Axiomatic Set Theory

This book is a basic reading on the Set Theory field. It works as an introductory and reference text, with a clear and concise style and a precise logic sequence on its exposition and development of contents. After reading it, I started to focus on more specific topics as Axiom of Choice and Banach-Tarski paradox, and I felt that it has prepared me to go further in my Math studies.

This is an excellent book for an undergraduate who has to begin doing abstract proof and has not had even rudimentary logic and symbolism.

Mathematics is a first order theory whose primitive formulae all take the form 'a is a member of b'. 'a' can be a set or atom; 'b' must be a set. If you do not object to the preceding sentence, then read on. Axiomatic Set Theory (AST) lays down the axioms of the now-canonical set theory due to Zermelo, Fraenkel (and Skolem), called ZFC. Building on ZFC, Suppes then derives the theory of cardinal and ordinal numbers, the integers, rationals, and reals, and the transfinite--Cantor's paradise. Suppes accomplishes in 250 well laid out pages what required 800 crabbed pages in Principia Mathematica. This book evolved out of a class Suppes taught at Stanford in the long ago 1950s. It has since remained the best book of its kind. The reason is that subsequent presentations of set theory are too difficult, too contrived, too clever by half. They disdain the basics as old hat. AST has several valuable pedagogical features. 1. The introduction to relations and functions is the best I know of. I am disappointed at how little attention has been devoted to relations and relational algebra in recent decades. 2. Suppes has a nice way of introducing a simple axiom, then showing that that axiom is a theorem when a more complicated axiom is later introduced. In particular, he develops the theory of cardinals by means of a temporary axiom to the effect that equipollent sets have identical cardinalities. This axiom becomes a theorem when the axiom of Choice is introduced in the final chapter. The axiom schema of Replacement is introduced as late as possible, to enable transfinite arithmetic. He then turns around and shows that Replacement makes Subsets and Pairing redundant. In my opinion, the greatest flaw of ZFC is that defining a cardinal number requires either the axiom of Choice, or Infinity plus the subtle notion of set rank. Frege and Russell had an appealing definition: a cardinal number is an equivalence class of sets under equipollence. That definition does not work in ZFC. It does work in Quinian set theory. Suppes does a yeoman's job of battling this flaw. 3. Suppes defines a finite set in the interesting way Tarski proposed in 1924. AST contains hundreds and hundreds of theorems, man of them useful classics. In many cases, the proof is an exercise. Suppes's proof are of the informal sort typical of mathematics. What AST does can be done more rigorously: type 'Metamath' into Google and see for yourself. Even though Suppes is a philosopher, this book is almost entirely a mathematical exercise. The reader will not get a good feel for how set theory is part of analytic philosophy, and how it has been a contentious subject. The writings of Fraenkel and Bar Hillel are better in these respects. Suppes does highlight the reservations re the axiom of Choice, but Cohen's proof that Choice is independent of ZF has largely laid those reservations to rest, except for those of us with constructive sympathies. AST gives no hint that Replacement and Power S

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 d

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.