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Produktbeschreibungen
Kurzbeschreibung
This book is intended to provide an account of those parts of contemporary set theory that are relevant to other areas of pure mathematics. Intended for advanced undergraduates and beginning graduate students, the text is written in an easy-going style, with a minimum of formalism. The book begins with a review of "naive" set theory; it then develops the Zermelo-Fraenkel axioms of the theory, showing how they arise naturally from a rigorous answer to the question, "what is a set?" After discussing the ordinal and cardinal numbers, the book then delves into contemporary set theory, covering such topics as: the Borel hierarchy, stationary sets and regressive functions, and Lebesgue measure. Two chapters present an extension of the Zermelo-Fraenkel theory, discussing the axiom of constructibility and the question of provability in set theory. A final chapter presents an account of an alternative conception of set theory that has proved useful in computer science, the non-well-founded set theory of Peter Aczel. The author is a well-known mathematician and the editor of the "Computers in Mathematics" column in the AMS Notices and of FOCUS, the magazine published by the MAA.
-- Dieser Text bezieht sich auf eine vergriffene oder nicht verfügbare Ausgabe dieses Titels.
Synopsis
This book is intended to provide an account of those parts of contemporary set theory that are relevant to other areas of pure mathematics. Intended for advanced undergraduates and beginning graduate students, the text is written in an easy-going style, with a minimum of formalism. The book begins with a review of "naive" set theory; it then develops the Zermelo-Fraenkel axioms of the theory, showing how they arise naturally from a rigorous answer to the question, "what is a set?" After discussing the ordinal and cardinal numbers, the book then delves into contemporary set theory, covering such topics as: the Borel hierarchy, stationary sets and regressive functions, and Lebesgue measure. Two chapters present an extension of the Zermelo-Fraenkel theory, discussing the axiom of constructibility and the question of provability in set theory. A final chapter presents an account of an alternative conception of set theory that has proved useful in computer science, the non-well-founded set theory of Peter Aczel. The author is a well-known mathematician and the editor of the "Computers in Mathematics" column in the AMS Notices and of FOCUS, the magazine published by the MAA.
In diesem Buch
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Einleitungssatz
Zermelo-Fraenkel set theory, which forms the main topic of the book, is a rigorous theory, based on a precise set of axioms. Lesen Sie die erste Seite
Buchdeckel | Copyright | Inhaltsverzeichnis | Auszug | Stichwortverzeichnis | Rückseite
Zermelo-Fraenkel set theory, which forms the main topic of the book, is a rigorous theory, based on a precise set of axioms. Lesen Sie die erste Seite
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Buchdeckel | Copyright | Inhaltsverzeichnis | Auszug | Stichwortverzeichnis | Rückseite
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27 von 27 Kunden fanden die folgende Rezension hilfreich
Superb!
4. Juni 2005
Format:Gebundene Ausgabe
Keith Devlin is one of those rare research mathematicians who is able to make recent advances in mathematics understandable and interesting to those whose mathematical education is obsolete or incomplete. I'm in the former category, having done my graduate work in pure math 50 years ago; although I've tried to keep up, constraints of time and other obligations have made it difficult.
Most modern texts on set theory put the reader to sleep, either because they avoid the important parts ("Set Theory for Those who Don't Want to Know It") or because they employ a degree of formalism that is quite difficult to grasp ("Set Theory Derived by Pure Propositional Logic, Step by Step"). Devlin's book avoids both traps. He presents modern advanced material that illuminates the subject admirably, but is careful not to submerge the reader in overwhelming finicky details. His discussions of constructive set theory, of independence proofs in set theory, and of non-well-founded set theory, are the first ones I've seen that get me excited enough to put the book aside and start exploring some of the implications on my own.
If I search for anything about the book to criticize, I find only one very minor thing. The sequence of proofs that show "Zorn's Lemma", the Axiom of Choice, the well-ordering principle, "Tukey's Lemma", etc to be equivalent to one another as an addition to the traditional Zermolo-Frankel axioms would be clearer if prefaced by an intuitive discssion of why the various steps in the chain of reasoning "ought" to work as they do; such a discussion helped me a lot many years ago to internalize what's going on. But that comment is just a nit.
On the other extreme, having once, 30+ years ago, being forced by the exigencies of a real-world problem to blunder through the creation of my own version of fragments of non-well-founded set theory, it gives me much joy to see it exounded as a coherent mthematical topic.
I read and reread this book, and drag it off the shelf when it occurs to me to ponder on some aspect that I don't fully recall. There are a number of other books on topics in pure mathematics about which I feel the same way, but they are a tiny minority among the deluge of texts that will never be read by anyone who doesn't have to. It's obviously an excellent text for advanced undergraduates and beginning graduate students, but beyond that, I recommend it to anyone with a working knowledge of pure math whose knowledge of set theory is somewhat behind current knowledge.
In short, buy a copy!
Most modern texts on set theory put the reader to sleep, either because they avoid the important parts ("Set Theory for Those who Don't Want to Know It") or because they employ a degree of formalism that is quite difficult to grasp ("Set Theory Derived by Pure Propositional Logic, Step by Step"). Devlin's book avoids both traps. He presents modern advanced material that illuminates the subject admirably, but is careful not to submerge the reader in overwhelming finicky details. His discussions of constructive set theory, of independence proofs in set theory, and of non-well-founded set theory, are the first ones I've seen that get me excited enough to put the book aside and start exploring some of the implications on my own.
If I search for anything about the book to criticize, I find only one very minor thing. The sequence of proofs that show "Zorn's Lemma", the Axiom of Choice, the well-ordering principle, "Tukey's Lemma", etc to be equivalent to one another as an addition to the traditional Zermolo-Frankel axioms would be clearer if prefaced by an intuitive discssion of why the various steps in the chain of reasoning "ought" to work as they do; such a discussion helped me a lot many years ago to internalize what's going on. But that comment is just a nit.
On the other extreme, having once, 30+ years ago, being forced by the exigencies of a real-world problem to blunder through the creation of my own version of fragments of non-well-founded set theory, it gives me much joy to see it exounded as a coherent mthematical topic.
I read and reread this book, and drag it off the shelf when it occurs to me to ponder on some aspect that I don't fully recall. There are a number of other books on topics in pure mathematics about which I feel the same way, but they are a tiny minority among the deluge of texts that will never be read by anyone who doesn't have to. It's obviously an excellent text for advanced undergraduates and beginning graduate students, but beyond that, I recommend it to anyone with a working knowledge of pure math whose knowledge of set theory is somewhat behind current knowledge.
In short, buy a copy!
16 von 16 Kunden fanden die folgende Rezension hilfreich
At times not so easygoing, but indeed a joy to read ...
31. Oktober 2001
Format:Gebundene Ausgabe
Anyone seeking a deeper understanding of the foundations of the mathematics will benefit from reading this excellent book.
Despite considerably abstract (almost no concrete examples), this book was carefully conceived to guide the reader through some of the most exciting contemporary ideas on set theory. If I had to name a minus about this book, I would mention the lack of solutions to the problems posted by the author. This makes the book a little less suitable for self-study.
Despite considerably abstract (almost no concrete examples), this book was carefully conceived to guide the reader through some of the most exciting contemporary ideas on set theory. If I had to name a minus about this book, I would mention the lack of solutions to the problems posted by the author. This makes the book a little less suitable for self-study.
Nevertheless, this book was written with care and love for the subject.
12 von 14 Kunden fanden die folgende Rezension hilfreich
Too short on explanation
12. Juli 2005
Format:Gebundene Ausgabe
This text is intended for seniors or beginning grads. The first three of seven chapters form a very quick survey of naive set theory. Since it aims at a more advanced audience, it is not as explanatory as Enderton and the exercises assume more maturity. Chapters 4 - 7 survey some advanced topics that aren't part of the usual introductory set theory course. These chapters have no exercises.
The development lacks a lot in clarity, exercises have only cursory introduction, and the author tends to get ahead of himself, assuming material before introducing it. The text by Roitman is much better and is targeted at the same audience.
The development lacks a lot in clarity, exercises have only cursory introduction, and the author tends to get ahead of himself, assuming material before introducing it. The text by Roitman is much better and is targeted at the same audience.
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