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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addition apply argument assertion associative axiom belongs called cardinal Cartesian product choice collection comparable complete concept condition consequence consider consists construction contains continuation countable defined definition denoted described determined disjoint distinct domain easy element empty equal equivalent exactly example EXERCISE exists extension fact follows formulation function function f given hence immediate implies important included induction infinite initial segment instance intersection inverse latter least maps mathematical maximal means namely natural numbers necessary non-empty notation Note Observe obtained one-to-one correspondence ordered pairs ordinal number partially ordered set particular possible preceding precise predecessors principle proof proper subset prove range reason relation restriction result sense sentence sequence set theory similar singleton specification subset successor Suppose symbol theorem thing tion transfinite true union unique upper bound usually write