Naive Set Theory
Every mathematician agrees that every mathematician must know some set theory; the disagreement begins in trying to decide how much is some. This book contains my answer to that question. The purpose of the book is to tell the beginning student of advanced mathematics the basic set theoretic facts of life, and to do so with the minimum of philosophical discourse and logical formalism. The point of view throughout is that of a prospective mathematician anxious to study groups, or integrals, or manifolds. From this point of view the concepts and methods of this book are merely some of the standard mathematical tools; the expert specialist will find nothing new here. Scholarly bibliographical credits and references are out of place in a purely expository book such as this one. The student who gets interested in set theory for its own sake should know, however, that there is much more to the subject than there is in this book. One of the most beautiful sources of set-theoretic wisdom is still Hausdorff's Set theory. A recent and highly readable addition to the literature, with an extensive and up-to-date bibliography, is Axiomatic set theory by Suppes.
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A U B arithmetic assertion axiom of choice axiom of extension axiom of pairing axiom of specification axiom of substitution belongs called cardinal Cartesian product chain collection of sets commutative countable sets countably infinite defined definition denoted disjoint sets domain empty equal equivalence class equivalence relation example ExERCISE exists a set f maps fact finite set follows function f hence implies included index set initial segment determined intersection inverse least element maximal element means natural numbers necessary and sufficient non-empty subset notation one-to-one correspondence ordered pairs ordinal number equivalent ordinal sum partially ordered set power set preceding paragraph predecessors proof proper subset prove result sentence sequence of type set of ordered set theory set-theoretic similar singleton successor set sufficient condition Suppose symbol tion transfinite induction transfinite recursion union unique unordered pair upper bound words write X X Y