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Set Theory With an Introduction to Real Point Sets 1st Edition by Abhijit Dasgupta 9781461488538 1461488532

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Set Theory With an Introduction to Real Point Sets 1st Edition Abhijit Dasgupta (Auth.) Digital Instant Download

Author(s): Abhijit Dasgupta (auth.)
ISBN(s): 9781461488538, 1461488532
Edition: 1
File Details: PDF, 2.17 MB
Year: 2014
Language: english
SKU: EB-4679088 Category: Tag:

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ISBN 10: 1461488532

ISBN 13: 9781461488538

Author: Abhijit Dasgupta

What is a number? What is infinity? What is continuity? What is order? Answers to these fundamental questions obtained by late nineteenth-century mathematicians such as Dedekind and Cantor gave birth to set theory. This textbook presents classical set theory in an intuitive but concrete manner. To allow flexibility of topic selection in courses, the book is organized into four relatively independent parts with distinct mathematical flavors. Part I begins with the Dedekind–Peano axioms and ends with the construction of the real numbers. The core Cantor–Dedekind theory of cardinals, orders, and ordinals appears in Part II. Part III focuses on the real continuum. Finally, foundational issues and formal axioms are introduced in Part IV. Each part ends with a postscript chapter discussing topics beyond the scope of the main text, ranging from philosophical remarks to glimpses into landmark results of modern set theory such as the resolution of Lusin’s problems on projective sets using determinacy of infinite games and large cardinals. Separating the metamathematical issues into an optional fourth part at the end makes this textbook suitable for students interested in any field of mathematics, not just for those planning to specialize in logic or foundations. There is enough material in the text for a year-long course at the upper-undergraduate level. For shorter one-semester or one-quarter courses, a variety of arrangements of topics are possible. The book will be a useful resource for both experts working in a relevant or adjacent area and beginners wanting to learn set theory via self-study.

Table of contents:

Part 1. Preliminaries: Sets, Relations, and Functions
Chapter 1 Dedekind: Numbers
Chapter 2 The Dedekind–Peano Axioms
Chapter 3 Dedekind’s Theory of the Continuum
Chapter 4 Postscript I: What Exactly Are the Natural Numbers?

Part 2. Cantor: Cardinals, Order, and Ordinals
Chapter 5 Cardinals: Finite, Countable, and Uncountable
Chapter 6 Cardinal Arithmetic and the Cantor Set
Chapter 7 Orders and Order Types
Chapter 8 Dense and Complete Orders
Chapter 9 Well-Orders and Ordinals
Chapter 10 Alephs, Cofinality, and the Axiom of Choice
Chapter 11 Posets, Zorn’s Lemma, Ranks, and Trees
Chapter 12 Postscript II: Infinitary Combinatorics

Part 3. Real Point Sets
Chapter 13 Interval Trees and Generalized Cantor Sets
Chapter 14 Real Sets and Functions
Chapter 15 The Heine–Borel and Baire Category Theorems
Chapter 16 Cantor–Bendixson Analysis of Countable Closed Sets
Chapter 17 Brouwer’s Theorem and Sierpinski’s Theorem
Chapter 18 Borel and Analytic Sets
Chapter 19 Postscript III: Measurability and Projective Sets

Part 4. Paradoxes and Axioms
Chapter 20 Paradoxes and Resolutions
Chapter 21 Zermelo–Fraenkel System and von Neumann Ordinals
Chapter 22 Postscript IV: Landmarks of Modern Set Theory

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Tags: Abhijit Dasgupta, Theory, Introductio, Real