Set Theory

• 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.

From the book reviews: "This book is an excellent introduction to set theory. Dasgupta (Univ. of Detroit Mercy) promotes reader/student interaction by integrating problems throughout the text instead of just providing occasional exercise sets. ... The book contains more than 630 frequently challenging exercises that will interest both upper-division students and readers with strong mathematical backgrounds. Summing Up: Highly Recommended. Upper-division undergraduates and above." (D. P. Turner, Choice, Vol. 52 (6), February, 2015) "This undergraduate textbook provides a thorough examination of the cardinals, ordinals, and the continuum. ... This work is a good introduction and would serve for two semesters of upper undergraduate study. ... Each part ends with remarks that are a departure point for further exploration. ... 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, MAA Reviews, November, 2014) "The present undergraduate textbook develops the core material on cardinals, ordinals, and the real line in an informal, predominantly intuitive but nevertheless concrete and rigorous manner. ... this lucidly written undergraduate set theory textbook is a welcome addition to the relevant literature, with many individual features and a remarkably high degree of thematic versatility." (Werner Kleinert, zbMATH, Vol. 1286 (2), 2014)