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 ...
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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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Until reading Halmos's "Naive Set Theory", I thought of this topic as one more branch of mathematics, akin to algebra, analysis, number theory, etc. I now understand that there's an alternative perspective: set theory constitutes the totality of mathematics, i.e., every branch of mathematics can be defined and developed within the set theoretic framework.
"Naive Set Theory" develops this idea in an "elementary" fashion: the exposition is self-containd and the proofs are short and straightforward. This does not mean that it's easy reading. But, the difficulties stem from the unusual notation, and can be overcome. Or, the book can be read for its ideas and its conclusions, which are presented with great clarity; Halmos is an excellent writer.
To whet your appetite for the book, I'll mention one fascinating result, which I've never come across before, despite having read a few other books on the subject of transfinte cardinal and ordinal numbers. The result is short and sweet. It's this: the cardinals are a subset of the ordinals.
