Set Theory With an Introduction to Real Point Sets

Set Theory With an Introduction to Real Point Sets / by 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 a...

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Bibliographic Details
Main Author: Dasgupta, Abhijit (Author)
Corporate Author: SpringerLink (Online service)
Format: eBook
Language:English
Published: New York, NY : Springer New York : Imprint: Birkhäuser, 2014.
Edition:1st ed. 2014.
Series:Springer eBook Collection.
Subjects:
Mathematical logic.
Mathematical analysis.
Analysis (Mathematics).
Algebra.
Topology.
Discrete mathematics.
Logic.
Electronic resources (E-books)
Online Access:Click to view e-book
Holy Cross Note:Loaded electronically.
Electronic access restricted to members of the Holy Cross Community.
Table of Contents:
  • 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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