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The Axiom of Choice (Dover Books on Mathematics) (Englisch) Taschenbuch – 24. Juli 2008

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  • Taschenbuch: 202 Seiten
  • Verlag: Dover Pubn Inc; Auflage: Dover. (24. Juli 2008)
  • Sprache: Englisch
  • ISBN-10: 0486466248
  • ISBN-13: 978-0486466248
  • Größe und/oder Gewicht: 21,1 x 13,7 x 1,3 cm
  • Durchschnittliche Kundenbewertung: 1.0 von 5 Sternen  Alle Rezensionen anzeigen (1 Kundenrezension)
  • Amazon Bestseller-Rang: Nr. 274.219 in Fremdsprachige Bücher (Siehe Top 100 in Fremdsprachige Bücher)

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2 von 31 Kunden fanden die folgende Rezension hilfreich Von Maximilian Oberbauer am 3. November 2008
Format: Taschenbuch
Ok. If i would have known what we was talking about, i would have had a chance to understand what he was talking about.
no chance.
no attempt to comunicate with readers.
zero didactic value.
when, oh when will they learn that writing book is about communication? --- with r e a d e r s?
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Die hilfreichsten Kundenrezensionen auf (beta) 3 Rezensionen
33 von 35 Kunden fanden die folgende Rezension hilfreich
Starts out a little rough 10. Mai 2010
Von Christopher Grant - Veröffentlicht auf
Format: Taschenbuch
(1) The second inequality on page 3, while true, is apparently only accidentally so, because the set on the right is not a subset of the set on the left. Changing [0,1] to [0,2] fixes this.

(2) The first reference to "rotations" on page 5 needs to be "nontrivial rotations"; otherwise, Q is the entire sphere, rather than a countable subset. (If you insist that rotations are automatically nontrivial, then the claim on page 4 that they can form a group is false.)

(3) X^i on the fourth line of the proof of Lemma 1.5 needs to be X_1 instead.

(4) The proof of Theorem 1.2 uses S to stand for two different things: a sphere and a countable subset thereof.

(5) The point p is supposed to be an arbitrary point in S's cone but outside of C's cone. If the S here is supposed to be the uncountable S, then this doesn't guarantee that the right-hand side of (1.7) is a disjoint union as apparently intended, but if the S here is the countable S, then we're trying to choose p from an empty set. I'm guessing that the countable S is intended, and that p is supposed to be in C's cone but outside S's cone, not the other way around.

These errors occur in the first ten pages of this book. Based on this small sample size, I'm finding it difficult to motivate myself to read any further.

Sometimes buying an inexpensive, uncorrected Dover reprint isn't such a bargain.
4 von 4 Kunden fanden die folgende Rezension hilfreich
Insightful presentation of AC issues and ZF models. 28. Januar 2013
Von Alan U. Kennington - Veröffentlicht auf
Format: Taschenbuch Verifizierter Kauf
This book is ideal if, like me, you are undecided about whether the axiom(s) of choice should be accepted, rejected, or carefully interpreted and managed. Jech summarises the relevant model theory and applies this to the principal AC issues.

This Dover book, "The axiom of choice", by Thomas Jech (ISBN 978-0-486-46624-8), written in 1973, should not be judged as a textbook on mathematical logic or model theory. It is clearly a monograph focused on axiom-of-choice questions. However, it contains many insights into mathematical logic and model theory which I have not obtained from the other 35 books on these subjects which I have on my book-shelf.

* Chapter 1 (8 pages): Introduction. Gives some really good reasons why you should reject the axiom of choice, especially Lebesgue non-measurable sets and a "paradoxical decomposition of the sphere", attributed to Hausdorff (1915) and Banach/Tarski (1924). The Banach-Tarski theorem is also mentioned (very briefly) by Roger Penrose in his popular book The Road to Reality, page 366, as a reason to reject AC.

* Chapter 2 (22 pages): AC equivalents and applications to get some "nice" theorems in many areas of mathematics, particularly the prime ideal theorem, applications of countable choice to analysis, and consequences for cardinal numbers. (In my opinion, AC is a magic wand which makes your wishes come true, not serious mathematics! But I'm in a minority on this issue.)

* Chapter 3 (13 pages): Consistency of AC with ZF, which is demonstrated by the constructible universe. (There's a very much better presentation of the constructible universe in Shoenfield's Mathematical Logic.) Also in this chapter is the definition of transitive models, particularly in relation to ZF and the 8 Gödel operations.

* Chapter 4 (11 pages): Permutation models, including the "basic Fraenkel model" (N1), the "second Fraenkel model" (N2), and the "ordered Mostowski model" (N3). The numbers in parentheses are my guess of the model number in the comprehensive book of ZF models, Consequences of the Axiom of Choice by Howard and Rubin.

* Chapter 5 (30 pages): Independence of the axiom of choice (from the ZF axioms). This goes further into model theory and describes the "basic Cohen model" (M1) and the "second Cohen model" (M7). (Howard/Rubin model number guesses in parentheses.) Various independence results follow from these models, including the 1963 Cohen proof that AC and GCH are independent of ZF.

* Chapter 6 (12 pages): Embedding theorems. Not of much interest to me personally.

* Chapter 7 (22 pages): Models with finite supports. Covers independence of AC from the Prime Ideal Theorem, independence of PIT from the Ordering Principle, and a couple of independence theorems for AC for finite sets.

* Chapter 8 (24 pages): Weaker versions of AC, including independence results for dependent choice.

* Chapter 9 (8 pages): Nontransferable statements. Some results using ZFA (ZF with atoms). I have no idea what this is about.

* Chapter 10 (10 pages): What you lose if you reject AC. This is a short summary, but it usefully includes the Solovay model (1965, 1970) which I'm pretty sure is model M5(ℵ) in Howard/Rubin. The Solovay model is interesting because dependent choice and countable choice hold, but general AC does not. (The Howard/Rubin book is totally comprehensive regarding consequences of rejecting AC and is therefore far preferable.)

* Chapter 11 (16 pages): What happens to cardinal number theory if you reject AC.

* Chapter 12 (16 pages): "Some properties contradicting the axiom of choice". Has something to do with measurable cardinals. Not of much interest to me (probably).

This book has lots of exercises, which I have not attempted because I'm too lazy. Maybe as much as a quarter or third of the book consists of exercises.

* Conclusion:
In my opinion, this is an invaluable book for its coverage of the basic issues concerning the axiom of choice. It is also particularly valuable for its clear (and brief) explanation of several ZF models which are particularly useful for demonstrating various independence and consistency results. The quite detailed summary of the Hausdorff/Banach/Tarski paradoxical decomposition of the sphere particularly interested me.

* PS. 2013-3-31.
Six days ago, I received Zermelo's Axiom of Choice: Its Origins, Development, and Influence (Dover Books on Mathematics). I haven't read enough of this book by Gregory H. Moore to write a review yet, but I am enormously impressed by it. I think it covers pretty much the same areas as the Jech book, and much, much more.
7 von 19 Kunden fanden die folgende Rezension hilfreich
Axiom of Choice 7. Oktober 2009
Von Dennis K. Morgan - Veröffentlicht auf
Format: Taschenbuch Verifizierter Kauf
Good book if you are trying to understand where the field of Mathematics is going: from Set Theory (Cantor & Godel), to modern math going forward. High School and College math skills necessary to grasp hard concepts. However, a good read for lay Mathematicians.
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