## Basic Set TheoryThe main notions of set theory (cardinals, ordinals, transfinite induction) are fundamental to all mathematicians, not only to those who specialize in mathematical logic or set-theoretic topology. Basic set theory is generally given a brief overview in courses on analysis, algebra, or topology, even though it is sufficiently important, interesting, and simple to merit its own dedicated treatment. This book provides just that in the form of a leisurely exposition for a diversified audience. It is suitable for a broad range of readers, from undergraduate students to professional mathematicians who want to finally find out what transfinite induction is and why it is always replaced by Zorn's Lemma. The text introduces all main subjects of ``naive'' (nonaxiomatic) set theory: functions, cardinalities, ordered and well-ordered sets, transfinite induction and its applications, ordinals, and operations on ordinals. Included are discussions and proofs of the Cantor-Bernstein Theorem, Cantor's diagonal method, Zorn's Lemma, Zermelo's Theorem, and Hamel bases. With over 150 problems, the book is a complete and accessible introduction to the subject. |

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algebraic argument assume axiom of choice axiomatic set theory base change belongs bijection binary Borel sets called Cantor Cantor–Bernstein Theorem chain coincides Consider the set construct contains continuum cardinality countable ordinals countable sets countable union decreasing sequence defined denoted disjoint equal example exceed the cardinality exists a one-to-one finite or countable finite set function f greatest element Hamel basis Hint implies infinite sequences infinite set intersection isomorphic least element least upper bound Let us prove linear combination linear space linearly independent set linearly ordered set mapping f mathematician maximal element minimal natural numbers nonnegative integers notation Note one-to-one correspondence order type order-isomorphic partially ordered set points poset Problem Proof properties pseudo-volume rank rational numbers real numbers recursive definition sequences of zeros smaller ordinals sound fragment statement subset transfinite induction tree uncountable unique values well-founded well-ordered set Zorn’s Lemma αβ