Notes on Set Theory

The axiomatic theory of sets is a vibrant part of pure mathematics, with its own basic notions, fundamental results, and deep open problems. It is also viewed as a foundation of mathematics so that "to make a notion precise" simply means "to define it in set theory." This book gives a solid introduction to "pure set theory" through transfinite recursion and the construction of the cumulative hierarchy of sets, and also attempts to explain how mathematical objects can be faithfully modeled within the universe of sets. In this new edition the author has added solutions to the exercises, and rearranged and reworked the text to improve the presentation.

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About the First Edition: This is a sophisticated undergraduate set theory text, brimming with mathematics, and packed with elegant proofs, historical explanations, and enlightening exercises, all presented at just the right level for a first course in set theory. - Joel David Hamkins, Journal of Symbolic Logic This is an excellent introduction to axiomatic set theory, viewed both as a foundation of mathematics and as a branch of mathematics with its own subject matter, basic results, open problems. - Achille C. Varzi, History and Philosophy of Logic

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Introduction.- Equinumerosity.- Paradoxes and axioms.- Are sets all there is?.- The natural numbers.- Fixed points.- Well ordered sets.- Choices.- Choice?s consequences.- Baire space.- Replacement and other axioms.- Ordinal numbers.- A. The real numbers.- B. Axioms and universes.- Index.

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