Axiomatic Set Theory

by Patrick C. Suppes

One of the most pressingproblems of mathematics over the last hundred years has been the What is a number? One of the most impressive answers has been the axiomatic development of set theory. The question raised "Exactly what assumptions, beyond those of elementary logic, are required as a basis for modern mathematics?" Answering this question by means of the Zermelo-Fraenkel system, Professor Suppes' coverage is the best treatment of axiomatic set theory for the mathematics student on the upper undergraduate or graduate level. The opening chapter covers the basic paradoxes and the history of set theory and provides a motivation for the study. The second and third chapters cover the basic definitions and axioms and the theory of relations and functions. Beginning with the fourth chapter, equipollence, finite sets and cardinal numbers are dealt with. Chapter five continues the development with finite ordinals and denumerable sets. Chapter six, on rational numbers and real numbers, has been arranged so that it can be omitted without loss of continuity. In chapter seven, transfinite induction and ordinal arithmetic are introduced and the system of axioms is revised. The final chapter deals with the axiom of choice. Throughout, emphasis is on axioms and theorems; proofs are informal. Exercises supplement the text. Much coverage is given to intuitive ideas as well as to comparative development of other systems of set theory. Although a degree of mathematical sophistication is necessary, especially for the final two chapters, no previous work in mathematical logic or set theory is required. For the student of mathematics, set theory is necessary for the proper understanding of the foundations of mathematics. Professor Suppes in Axiomatic Set Theory provides a very clear and well-developed approach. For those with more than a classroom interest in set theory, the historical references and the coverage of the rationale behind the axioms will provide a strong background to the major developments in the field. 1960 edition.

Genres: Mathematics, Philosophy, Logic, Science, Nonfiction, Textbooks

356 pages, Kindle Edition

First published January 1, 1968

Reviews

[Joseph Kulisics · Rating 5 out of 5]

I worked through all of the problems in the book. Based on my cursory assessment of other treatments of the subject, I think that the book gives one of the best treatments of axiomatic set theory that I've seen. I saw another reviewer pay special attention to the fact that the book uses a simplifying assumption, a special axiom for cardinals, to develop a cardinal arithmetic, which would ordinarily be much more complicated in its initial stages than the comparable development of ordinal arithmetic. I believe that the author later justifies the axiom with the Axiom of Choice. While a development of cardinal arithmetic independent of the Axiom of Choice and based on the rank of a set might be more intellectually parsimonious, since the Axiom of Choice is necessary to prove basic claims like the claim that the product of an infinite collection of non-empty sets is non-empty, as far as I'm concerned, the development is sufficiently rigorous and satisfying.

[Stephen · Rating 4 out of 5]

Unfortunately, many mathematics degree programs introduce set theory when convenient. This book fills in any holes that may still remain after going through your classes.

[Luca Prosperi · Rating 5 out of 5]

Spectacular reference book. All in one go...will require several detailed visits to specific chapters.

[Rick · Rating 4 out of 5]

If you want to learn Set Theory, go for a more modern treatment. If you already know Set Theory, but are interested in its history, then this is a great read.

[Mark6712]

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