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Customer Reviews (2)

Starts out a little rough
(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.

Axiom of Choice
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. ... Read more

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AC, the axiom of choice, because of its non-constructive character, is the most controversial mathematical axiom, shunned by some, used indiscriminately by others. This treatise shows paradigmatically that:

- Disasters happen without AC: Many fundamental mathematical results fail (being equivalent in ZF to AC or to some weak form of AC).

- Disasters happen with AC: Many undesirable mathematical monsters are being created (e.g., non measurable sets and undeterminate games).

- Some beautiful mathematical theorems hold only if AC is replaced by some alternative axiom, contradicting AC (e.g., by AD, the axiom of determinateness).

Illuminating examples are drawn from diverse areas of mathematics, particularly from general topology, but also from algebra, order theory, elementary analysis, measure theory, game theory, and graph theory.

Customer Reviews (1)

An essential resource for this important Axiom of Mathematics
As a graduate student in Mathematics, I felt self-assured that my knowledge in mathematics was complete enough to consider myself competent. This book showed me how wrong I was. Even though you are taught to notice when the axiom of choice is used, at least I never thought much about it. Even though I don't consider myself a fan of set theory, I think it is important to see these details at least once in your life. Anyway, give this book a chance if you can borrow it from your local library. You won't regret it, I can assure that. ... Read more

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This book presents an overview of the development of theAxiom of Choice since its introduction by Zermelo at thebeginning of the last century. The book surveys the Axiom ofChoice from three perspectives. The first, or mathematicalperspective, is that of the "working mathematician". Thisperspective brings into view the manifold applications of theAxiom of Choice-usually in the guise of Zorn`s Lemma- in agreat variety of areas of mathematics. The second,foundational, perspective is that of the logician orconstructive mathematician concerned with the foundationalstatus of the Axiom of Choice. The third, topos-theoretical,perspective is that taken by the mathematician or logicianinvestigating the role of the Axiom of Choice in topos theory.Certain topics-for instance mathematical applications of theAxiom, and its relationship with logic-are discussed inconsiderable detail. Others-notably the consistency andindependence of the Axiom of the usual systems of settheory-are given no more than summary treatment, thejustification here being that these topics have been given fullexpositions elsewhere.It is hoped that the book will be of interest to logicians andmathematicians, both professional and prospective. ... Read more

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This monograph contains a selection of over 250 propositions which are equivalent to AC. The first part on set forms has sections on the well-ordering theorem, variants of AC, the law of the trichotomy, maximal principles, statements related to the axiom of foundation, forms from algebra, cardinal number theory, and a final section of forms from topology, analysis and logic. The second part deals with the axiom of choice for classes - well-ordering theorem, choice and maximal principles. ... Read more

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This book, Consequences of the Axiom of Choice, is a comprehensive listing of statements that have been proved in the last 100 years using the axiom of choice. Each consequence, also referred to as a form of the axiom of choice, is assigned a number. Part I is a listing of the forms by number. In this part each form is given together with a listing of all statements known to be equivalent to it (equivalent in set theory without the axiom of choice). In Part II the forms are arranged by topic. In Part III we describe the models of set theory which are used to show non-implications between forms. Part IV, the notes section, contains definitions, summaries of important sub-areas and proofs that are not readily available elsewhere. Part V gives references for the relationships between forms and Part VI is the bibliography. Part VII is contained on the floppy disk which is enclosed in the book. It contains a table with form numbers as row and column headings. The entry in the table in row $n$, column $k$ gives the status of the implication ``form $n$ implies form $k$''. Software for easily extracting information from the table is also provided. Features: complete summary of all the work done in the last 100 years on statements that are weaker than the axiom of choice software provided gives complete, convenient access to information about relationships between the various consequences of the axiom of choice and about the models of set theory descriptions of more than 100 models used in the study of the axiom of choice an extensive bibliography About the software: Tables 1 and 2 are accessible on the PC-compatible software included with the book. In addition, the program maketex.c in the software package will create TeX files containing copies of Table 1 and Table 2 which may then be printed. (Tables 1 and 2 are also available at the authors' Web sites: http://www.math.purdue.edu/$\sim$jer/ or http://www.emunix.emich.edu/$\sim$phoward/.) Detailed instructions for setting up and using the software are included in the book's Introduction, and technical support is available directly from the authors. ... Read more

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Purchase includes free access to book updates online and a free trial membership in the publisher's book club where you can select from more than a million books without charge. Chapters: Axiom of Choice, Zorn's Lemma, Well-Ordering Theorem, Tychonoff's Theorem, Hausdorff Maximal Principle, König's Lemma, C-K Theory, Boolean Prime Ideal Theorem, König's Theorem, Szpilrajn Extension Theorem, Axiom of Countable Choice, Choice Function, Uniformization, Axiom of Dependent Choice, Tukey's Lemma. Excerpt:In mathematics , the axiom of choice , or AC , is an axiom of set theory . Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin, even if there are infinitely many bins and there is no "rule" for which object to pick from each. The axiom of choice is not required if the number of bins is finite or if such a selection "rule" is available. The axiom of choice was formulated in 1904 by Ernst Zermelo . Although originally controversial, it is now used without reservation by most mathematicians. One motivation for this use is that a number of important mathematical results, such as Tychonoff's theorem , require the axiom of choice for their proofs. Contemporary set theorists also study axioms that are not compatible with the axiom of choice, such as the axiom of determinacy . Unlike the axiom of choice, these alternatives are not ordinarily proposed as axioms for mathematics, but only as principles in set theory with interesting consequences. Statement A choice function is a function f , defined on a collection X of nonempty sets, such that for every set s in X , f ( s ) is an element of s . With this concept, the axiom can be stated: For any set X of nonempty sets, there exists a choice function f defined on X . Thus the negation of the axiom of choice states that there exists a set of nonempty sets which has no choice function. Each choice fu... ... Read more

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Chapters: Mastan Ensemble, the Kamkars, Afsaneh Ballet, Chemirani Ensemble, Lian Ensemble, Axiom of Choice, Masters of Persian Music, Shams Ensemble, Dastan Ensemble, Nour Ensemble, Hamavayan Ensemble, Ghazal, Aref Ensemble, Zoufonoun Ensemble. Source: Wikipedia. Pages: 45. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: The Mastn Ensemble (Persian: - Gorouh-e Mastn) is an Iranian musical group performing traditional Persian music. The group was founded as Hafez Ensemble in 2005 by Mr Sa'id Jafarzdeh Homy ( ), also known as Parvz Homy ( ) and Homy Foomani. The group's performance follows a Persian tradition of music making known as Chmeh Sar'i ( ) according to which one person is in charge of writing the poems, composing the musics, arranging the instruments as well as singing the songs. The instruments employed by this Ensemble include Daf, Dayereh, Kamncheh, Lute, Reed, Robab, Santour, Tr and Tonbak. The Mastn Ensemble derive their inspirations from such Sufi poets as Farid ad-Din Attar, Hafez and Rumi. The performances of this Ensemble are emotionally intense and the thoughts and meanings conveyed by their musics and songs are both philosophical and contemplative. The composition of the ensemble is subject to some variations at different performances. Ney: Pasha Hanjani Kamancheh: Sina Jahanabadi Tombak: Arjang Faramarzi Santour: Mahmoud Nozari Tar: Azad Mirzapour Oud: Ali Pajouheshgar Kamancheh: Alireza Mehdizadeh ...More: http://booksllc.net/?id=18578496 ... Read more

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In mathematics, the constructible universe (or Gödel'sconstructible universe), denoted L, is a particular classof sets which can be described entirely in terms of simplersets. It was introduced by Kurt Gödel in his 1938paper "The Consistency of the Axiom of Choice and of theGeneralized Continuum-Hypothesis". In this, he proved thatthe constructible universe is an inner model of ZF settheory, and also that the axiom of choice and thegeneralized continuum hypothesis are true in theconstructible universe. This shows that both propositionsare consistent with the basic axioms of set theory, if ZFitself is consistent. Since many other theorems only holdin systems in which one or both of the propositions istrue, their consistency is an important result. ... Read more

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Chapters: Mastan Ensemble, the Kamkars, Kahtmayan, Niyaz, Vas, Vaspooher, Quark Kent, Axiom of Choice, Masters of Persian Music, Abjeez, Dastan Ensemble, Nour Ensemble, Shahin and Sepehr, Ghazal, Saamet, Iranian Electronic Music Bands. Source: Wikipedia. Pages: 47. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: The Mastn Ensemble (Persian: - Gorouh-e Mastn) is an Iranian musical group performing traditional Persian music. The group was founded as Hafez Ensemble in 2005 by Mr Sa'id Jafarzdeh Homy ( ), also known as Parvz Homy ( ) and Homy Foomani. The group's performance follows a Persian tradition of music making known as Chmeh Sar'i ( ) according to which one person is in charge of writing the poems, composing the musics, arranging the instruments as well as singing the songs. The instruments employed by this Ensemble include Daf, Dayereh, Kamncheh, Lute, Reed, Robab, Santour, Tr and Tonbak. The Mastn Ensemble derive their inspirations from such Sufi poets as Farid ad-Din Attar, Hafez and Rumi. The performances of this Ensemble are emotionally intense and the thoughts and meanings conveyed by their musics and songs are both philosophical and contemplative. The composition of the ensemble is subject to some variations at different performances. Ney: Pasha Hanjani Kamancheh: Sina Jahanabadi Tombak: Arjang Faramarzi Santour: Mahmoud Nozari Tar: Azad Mirzapour Oud: Ali Pajouheshgar Kamancheh: Alireza Mehdizadeh ...More: http://booksllc.net/?id=18578496 ... Read more

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This digital document is a journal article from Mathematical Social Sciences, published by Elsevier in 2006. The article is delivered in HTML format and is available in your Amazon.com Media Library immediately after purchase. You can view it with any web browser.

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Under two reasonable axioms, a generalized variant of the Edgeworth-Pareto principle in terms of a choice function is substantiated. According to this principle, for any choice function, its value must be a subset of the Pareto set. It is shown that, if at least one of the axioms is ignored, then the Edgeworth-Pareto principle may be violated. As a particular case, the Edgeworth-Pareto principle in terms of preference relations is derived. ... Read more

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High Quality Content by WIKIPEDIA articles! Zorn's lemma, also known as the Kuratowski?Zorn lemma, is a proposition of set theory that states: Every partially ordered set, in which every chain (i.e. totally ordered subset) has an upper bound, contains at least one maximal element.It is named after the mathematicians Max Zorn and Kazimierz Kuratowski.The terms are defined as follows. Suppose (P,?) is a partially ordered set. A subset T is totally ordered if for any s, t in T we have either s ? t or t ? s. Such a set T has an upper bound u in P if t ? u for all t in T. Note that u is an element of P but need not be an element of T. A maximal element of P is an element m in P such that for no element x in P, m < x.Zorn's lemma is equivalent to the well-ordering theorem and the axiom of choice, in the sense that any one of them, together with the Zermelo?Fraenkel axioms of set theory, is sufficient to prove the others. ... Read more