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Publisher:

Dover Publications

Publication Date:

1950

Number of Pages:

144

Format:

Paperback

Price:

8.95

ISBN:

9780486601410

Category:

Monograph

[Reviewed by , on ]

Allen Stenger

09/5/2011

This text provides a very old-fashioned view of set theory, presenting it essentially as Georg Cantor created it, before it was axiomatized by Zermelo, Fraenkel, and others. It therefore deals almost exclusively with transfinite arithmetic and has little about what we today consider to be set theory. The present work was originally published in 1950 as an English translation of the 1947 German second edition. Some of the notation is archaic; for example, it uses the numeral 0 for the null set, and uses + and · for set union and intersection.

The book has lots of examples (although no exercises) and provides a very clear exposition of cardinals and ordinals (unusually, cardinals are developed first). The discussion of well-ordering is also good, although the proof of the well-ordering theorem follows Zermelo’s first proof from 1904, which is very complicated and gives the impression that it is an unconditional theorem instead of an equivalent form of the Axiom of Choice. In fact the book does not mention of the Axiom of Choice or of Zorn’s Lemma anywhere. The final chapter discusses some of the paradoxes that arose from Cantor’s formulation, without giving any solutions and without mentioning the axiomatizations that are intended to solve these.

Bottom line: A good book with few prerequisites for learning about transfinite arithmetic but is not a course in set theory.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.

INTRODUCTION | |||||||

CHAPTER I. THE RUDIMENTS OF SET THEORY | |||||||

1. A First Classification of Sets | |||||||

2. Three Remarkable Examples of Enumerable Sets | |||||||

3. "Subset, Sum, and Intersection of Sets; in Particular, of Enumerable Sets" | |||||||

4. An Example of a Nonenumerable Set | |||||||

CHAPTER II. ARBITRARY SETS AND THEIR CARDINAL NUMBERS | |||||||

1. Extensions of the Number Concept | |||||||

2. Equivalence of Sets | |||||||

3. Cardinal Numbers | |||||||

4. Introductory Remarks Concerning the Scale of Cardinal Numbers | |||||||

5. F. Bernstein's Equivalence-Theorem | |||||||

6. The Sum of Two Cardinal Numbers | |||||||

7. The Product of Two Cardinal Numbers | |||||||

8. The Sum of Arbitrarily Many Cardinal Numbers | |||||||

9. The Product of Arbitrarily Many Cardinal Numbers | |||||||

10. The Power | |||||||

11. Some Examples of the Evaluation of Powers | |||||||

CHAPTER III. ORDERED SETS AND THEIR ORDER TYPES | |||||||

1. Definition of Ordered Set | |||||||

2. Similarity and Order Type | |||||||

3. The Sum of Order Types | |||||||

4. The Product of Two Order Types | |||||||

5. Power of Type Classes | |||||||

6. Dense Sets | |||||||

7. Continuous Sets | |||||||

CHAPTER IV. WELL-ORDERED SETS AND THEIR ORDINAL NUMBERS | |||||||

1. Definition of Well-ordering and of Ordinal Number | |||||||

2. "Addition of Arbitrarily Many, and Multiplication of Two, Ordinal Numbers" | |||||||

3. Subsets and Similarity Mappings of Well-ordered Sets | |||||||

4. The Comparison of Ordinal Numbers | |||||||

5. Sequences of Ordinal Numbers | |||||||

6. Operating with Ordinal Numbers | |||||||

7. "The Sequence of Ordinal Numbers, and Transfinite Induction" | |||||||

8. The Product of Arbitrarily Many Ordinal Numbers | |||||||

9. Powers of Ordinal Numbers | |||||||

10. Polynomials in Ordinal Numbers | |||||||

11. The Well-ordering Theorem | |||||||

12. An Application of the Well-ordering Theorem | |||||||

13. The Well-ordering of Cardinal Numbers | |||||||

14. Further Rules of Operation for Cardinal Numbers. Order Type of Number Classes | |||||||

15. Ordinal Numbers and Sets of Points | |||||||

CONCLUDING REMARKS | |||||||

BIBLIOGRAPHY | |||||||

KEY TO SYMBOLS | |||||||

INDEX |

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