This classic by one of the 20th century's most prominent mathematicians offers a concise introduction to set theory. Suitable for advanced undergraduates and graduate students in mathematics, it employs the language and notation of informal mathematics. Topics include the basic concepts of set theory, cardinal numbers, transfinite methods, and a good deal more in 25 brief chapters.
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This classic by one of the 20th century's most prominent mathematicians offers a concise introduction to set theory. Suitable for advanced undergraduates and graduate students in mathematics, it employs the language and notation of informal mathematics. Topics include the basic concepts of set theory, cardinal numbers, transfinite methods, and a good deal more in 25 brief chapters.
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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.
