Customer Reviews

Theory of Sets (Dover Books on Mathematics)

5 star:		(4)
4 star:		(2)
3 star:		(0)
2 star:		(0)
1 star:		(1)

 

 

This is a very good introduction to Set theory. It starts with basics, and takes one through to The Axiom of Choice or well-ordering theorem. Set theory being a rather abstract subject, most books do not give a motivation for the various terminology in set theory (which enhances understanding of why many concepts had to be introduced. The book however uses strange archaic Math typography, because of which through out the book I kept on trying to figure out whether it is a srcipt pi or script R. This sometimes confuses the thought process. But overall, the book is quite easy to read and manages to convey to the reader some of the beauty of set theory. At points, one can't help but smile at the ingenuity and the audacity of human thought.

I recommened it highly.

Kamke, along with Suppes and Halmos, is one of the classic texts in the theory of sets. Kamke is a bit unique, I think, in that he brings up terms such as jump, cut, and gap. Those are currently out-of-fashion but are in fact more intuitive and give you a direct philosophical grasp. A must-read.

This is a great little book of math for all of those interested in the finer points of set theory. I'm an undergrad math major and found the material challenging yet rewarding. If you are as young as I am, or an experienced mathematician, I believe that this book has something for you to think about. From the first you'll be taken along at a steady pace covering theorems that are quite interesting.

I agree completely with RJ's review, including the his/her frustration with the typography in which capital letters in what I believe is German script are used to denominate sets. Still, once I had that sorted out this was a straightforward and rewarding read. (Other than the capital German letters used to denominate sets, the notation is relatively easy to follow as well, and contrary to another review, there is a perfectly good index to the places where each symbol is introduced.) I intend to supplement its intuitive approach by reading Suppes' presentation of axiomatic set theory.

A lovely old book, a classic, brief and clear introduction to set theory: cardinal numbers, well-ordering, ordinal numbers. Everyone should own a copy.

I purchased this on a recommendation by one of my professors. This book is readily accessible to undergraduates in mathematics and provides almost all the needs of the same. Though it does not delve deeply into the means of resolving the contradictions of `naive' set theory, it at least presents the paradoxes well. Those who take a more philosophical interest in mathematics can find something to appreciate here.

There is one major deficiency, however. Even at the undergraduate level, it is important to understand at least one aspect of axiomatic set theory: namely, the application of the axiom of choice (Zorn's lemma). This is especially relevant when studying ring theory. I might also add that the index, vital for the utility of such a work on the assumption of an undergraduate target audience, is quite sparse; so the value of the book as a sort of `pocket reference' is limited. The notation is also somewhat out of keeping with modern conventions, e.g. the use of `0' to denote the empty set, but nothing overly confusing. But undergraduates can take advantage of the work's concision - there are only 131 pages of meat - by simply working through it. The exercise is worth the time.

I would recommend this to any undergraduate in mathematics or in any field which depends heavily on mathematics. (One should be completely ready by sophomore or junior year. No analysis or abstract algebra is required.) The age of the work - it was originally published in 1950 - is actually an advantage for students working with older professors, and nothing has gone out of date. So long as readers supplement the work with study of Zorn's lemma, there is no need for more set theory until grad school. (Even then, only those within certain specialties of mathematics employ axiomatic set theory. So as far as the average mathematician-in-training is concerned, this should hold you over until graduate courses make the contrary obvious.)

A brief outline:

Chapter 1 presents basic introductory material, such as the definition of a naive set, the basic set-theoretic operations, and Cantor's diagonalization argument for the non-enumerability of the reals.

Chapter 2 defines equivalent sets (this is actually an axiom in ZFC) and equivalence relations, basic consequences of this equivalence, and introduces the cardinals and generalizes the operations of sum and product to transfinite cardinals. Again, the notation is unconventional. That the power set of a set is always of strictly greater cardinality than the set is also proven.

Chapter 3 concerns ordered sets. Similarity of orders (a morphic relation between ordered sets) is discussed, as are sums and products of orderings by type.

Chapter 4 concerns well-ordered sets, i.e. ordered sets with a minimal element. Products and sums of such ordered sets are discussed, as are similarity mappings between them. The full meaning of trichotomy in ordering is explained, and the well-ordering theorem is proved. Transfinite induction is also explained. Further results are relevant to topology, such as the closure of unions of closed sets.

The concluding remarks concern the paradoxes of naive set theory.

For $7 you get what you pay for; absolutely nothing. At no time does this book describe the notation inside and sporadically jumps from one incomplete, detailless proof to another. This book bearly ranks as a pocket reference. If there was an option for no stars, this book should get it.