Set Theory: With an Introduction to Real Point Sets

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.