Book 2014

Set Theory

With an Introduction to Real Point Sets

Authors:

ISBN: 978-1-4614-8853-8 (Print) 978-1-4614-8854-5 (Online)

Table of contents (22 chapters)

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  1. Front Matter

    Pages i-xv

  2. No Access

    Chapter

    Pages 1-23

    Preliminaries: Sets, Relations, and Functions

  3. Dedekind: Numbers

    1. Front Matter

      Pages 25-27

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      Chapter

      Pages 29-46

      The Dedekind–Peano Axioms

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      Chapter

      Pages 47-65

      Dedekind’s Theory of the Continuum

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      Chapter

      Pages 67-72

      Postscript I: What Exactly Are the Natural Numbers?

  4. Cantor: Cardinals, Order, and Ordinals

    1. Front Matter

      Pages 73-75

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      Chapter

      Pages 77-107

      Cardinals: Finite, Countable, and Uncountable

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      Chapter

      Pages 109-129

      Cardinal Arithmetic and the Cantor Set

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      Chapter

      Pages 131-147

      Orders and Order Types

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      Chapter

      Pages 149-174

      Dense and Complete Orders

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      Chapter

      Pages 175-198

      Well-Orders and Ordinals

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      Chapter

      Pages 199-219

      Alephs, Cofinality, and the Axiom of Choice

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      Chapter

      Pages 221-243

      Posets, Zorn’s Lemma, Ranks, and Trees

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      Chapter

      Pages 245-250

      Postscript II: Infinitary Combinatorics

  5. Real Point Sets

    1. Front Matter

      Pages 251-253

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      Chapter

      Pages 255-264

      Interval Trees and Generalized Cantor Sets

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      Chapter

      Pages 265-279

      Real Sets and Functions

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      Chapter

      Pages 281-299

      The Heine–Borel and Baire Category Theorems

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      Chapter

      Pages 301-311

      Cantor–Bendixson Analysis of Countable Closed Sets

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      Chapter

      Pages 313-319

      Brouwer’s Theorem and Sierpinski’s Theorem

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      Chapter

      Pages 321-343

      Borel and Analytic Sets

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      Chapter

      Pages 345-355

      Postscript III: Measurability and Projective Sets

  6. Paradoxes and Axioms

    1. Front Matter

      Pages 357-359

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      Chapter

      Pages 361-367

      Paradoxes and Resolutions

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