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Set Theory with an Introduction to Real Point Sets

Abhijit Dasgupta
Publisher: 
Birkhäuser
Publication Date: 
2014
Number of Pages: 
444
Price: 
59.99
ISBN: 
9781461488538
Category: 
Textbook
[Reviewed by
Tom Schulte
, on
11/13/2014
]

This undergraduate textbook provides a thorough examination of the cardinals, ordinals, and the continuum. The book offers breadth: Dedekind, Peano, Cantor, ZF set theory, etc. Depth comes from presenting each topic in a historical context, with motivations alternating with breakthroughs. The concluding section, on paradoxes and special axioms, is cogent and enlightening standalone reading. While it might seem odd to put the discussion of the paradoxes that stimulated the development of the theory segregated to a near-appendix, this fits the informal approach of the book, in which the author develops ideas often untethered by any specific axiom system. The result is a pace that moves briskly to connect ideas typically presented many chapters apart and allows at times a hint of enthusiasm to emerge, as in

…strangely enough, a one-to-one correspondence between the whole and the strictly smaller part is established by nn2, showing that the size of the part is equal to the size of the whole, not smaller!

Adverbs and exclamations rarely ornament set theory textbooks!

This work is a good introduction and would serve for two semesters of upper undergraduate study. It is also a concise companion to any assigned text, indeed one I wish I had had available when I learned this material.

The book has four distinct sections which may be read independently. Part I moves from the Dedekind–Peano axioms to develop the real numbers. The Cantor–Dedekind theory giving the taxonomy of cardinals, orders, and ordinals makes up Part II. Part III explores the real continuum through Cantor Sets, category theory, Heine-Borel, and more. A concluding section covers Zermelo-Fraenkel set theory, paradoxes, and more. Appendices give the ZF axioms, discuss Lebesgue Measure, and give proofs of the uncountability of the reals. Each part ends with remarks that are a departure point for further exploration. The text is richly seasoned with posed problems and proofs left to the reader. The author’s clear interest in the subject matter and economy of presentation makes this an effective tool for learning set theory in the lecture hall or through self-study.


Tom Schulte gives guided tours of the tamed and domesticated areas of the reals to students at Oakland Community College in Michigan.

1 Preliminaries: Sets, Relations, and Functions

Part I Dedekind: Numbers
2 The Dedekind–Peano Axioms
3 Dedekind’s Theory of the Continuum
4 Postscript I: What Exactly Are the Natural Numbers?

Part II Cantor: Cardinals, Order, and Ordinals
5 Cardinals: Finite, Countable, and Uncountable
6 Cardinal Arithmetic and the Cantor Set
7 Orders and Order Types
8 Dense and Complete Orders
9 Well-Orders and Ordinals
10 Alephs, Cofinality, and the Axiom of Choice
11 Posets, Zorn’s Lemma, Ranks, and Trees
12 Postscript II: Infinitary Combinatorics

Part III Real Point Sets
13 Interval Trees and Generalized Cantor Sets
14 Real Sets and Functions
15 The Heine–Borel and Baire Category Theorems
16 Cantor–Bendixson Analysis of Countable Closed Sets
17 Brouwer’s Theorem and Sierpinski’s Theorem
18 Borel and Analytic Sets
19 Postscript III: Measurability and Projective Sets

Part IV Paradoxes and Axioms
20 Paradoxes and Resolutions
21 Zermelo–Fraenkel System and von Neumann Ordinals
22 Postscript IV: Landmarks of Modern Set Theory

Appendices
A Proofs of Uncountability of the Reals
B Existence of Lebesgue Measure
C List of ZF Axioms

References

List of Symbols and Notations

Index

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