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Customer Reviews (5)

Set Theory and the Continuum Hypothesis Review
It is a book that the most part of him is written in a naive form(not in formal logic).
You need a basic knowledge of Set Theory(like Halmos Book).
Very interesting and the book started from the root of the problem.
Very Good

Definitive and Brilliant
This is still the definitive work on set theory and the continuum hypothesis. Although extremely terse, it is wonderfully clear and unburdened by the technical and pedantic details that doom many books in the subject. If you cannot track this down right now be patient, the American Mathematical Society is going to be reprinting it.

Professor Cohen passed away in March of 2007, but thankfully this book remains as a testament to his genius. Originally trained as an analyst, he began working on the continuum hypothesis knowing almost nothing about logic or set theory. Within two years he mastered the subject and solved the greatest outstanding problem in the field (and arguably in all of mathematics). Read this book if you want to understand one of the deepest ideas in all of human thought.

All-time classic -- a "desert island book"
Paul Cohen's "Set Theory and the Continuum Hypothesis" is not only the best technical treatment of his solution to the most notorious unsolved problem in mathematics, it is the best introduction to mathematical logic (though Manin's "A Course in Mathematical Logic" is also remarkably excellent and is the first book to read after this one).

Although it is only 154 pages, it is remarkably wide-ranging, and has held up very well in the 37 years since it was first published.Cohen is a very good mathematical writer and his arrangement of the material is irreproachable.All the arguments are well-motivated, the number of details left to the reader is not too large, and everything is set in a clear philosophical context. The book is completely self-contained and is rich with hints and ideas that will lead the reader to further work in mathematical logic.

It is one of my two favorite math books (the other being Conway's "On Numbers and Games").My copy is falling apart from extreme overuse.

A priceless gem
An "older" original work of a great mathematican.Not the best book to read about the subject, but certainly a collectors item.
I happen to have one because my master thesis used it as a major reference.It's the one book that everyone interested in the foundations of mathematics should own - if you can still get one.

Brilliant example of human greatness
This is one of the greatest math works ever. Prof. Cohen solves one of the most important problems in mathematics of this century, and this book explains his thinking and methods. A must for anyone who is reallyinterested in mathematics. ... Read more

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Customer Reviews (3)

The symbols ARE missing
Perhaps there were different printing runs, but as the first reviewer experienced, my copy lacked the symbols for membership, subset, quantifiers, etc. It really does make the book useless. I too would like to have a correctly printed copy.
Update. I returned the book, explaining that the symbols were missing and got a replacement yesterday, Sept. 29. Unfortunately the replacement also lacks the symbols. The reviewer who had symbols is in Europe. Perhaps the US printing is messed up. I am going to go to a book store where I can look at the pages before I buy the book.

A fair review of an original book on set theory
Apparently some people had bought a printing of this Dover edition without maths symbols. In my copy of this Dover edition all maths symbols are here. Maybe there are different print runs and the first is without maths symbol. Maybe I have bought (with Amazon.fr) a second printing where the problem has been corrected.

Turning to the content this book is on independence proofs in set theory.

The first part, probably written by Smullyan (his witty style is easily recognized), begins with an exposition of the basics of set theory (with an axiomatics based on the concepts of sets and classes) and then exposes the consistency proofs of Gôdel of the axiom of choice and of the generalized continuum hypothesis. It is very readable, if not always as precise as it should be necessary.

The second part is the most original. It exposes the method of forcing with the help of a modicum of modal logic (all the necessary concepts of modal logic are explained), then proceed to prove the Independence of the Axiom of Choice and of the Continuum Hypothesis. The use of modal logic helps very much to understand the forcing method. But I must say that some points at the beginning of the exposition contain errors or are not exposed with all necessary details. And many points are left as exercices (from easy to difficult) to the reader. This is why I give only 3 stars to the book.

But overall, it is a very easy and readable introduction to the independence proofs in set theory, only to be compared with Kunen's "Set Theory".

Unreadable due to missing symbols
I just purchased the book.It is unreadable as many of the mathematical symbols are missing.The introduction makes it sound very interesting. ... Read more

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The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory by Kurt GödelKurt Gödel (1906-1978) was a logician, mathematician and philosopher. He is regarded as one of the most significant logicians of all time, who's work has had immense impact upon scientific and philosophical thinking in the 20th century.Gödel is best known for his three theorems, which set forth and explain the foundations of mathematics.This book provides the proof for the third of his three theorems.Included is a new foreword by Richard Laver, Professor of Mathematics at the University of Colorado at Boulder, who explains in simple, non-technical terms the basic, underlying ideas behind Gödel's theorems and proofs and why they are relevant and important. ... Read more

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Chapters: Linguistic Relativity, Continuum Hypothesis, Documentary Hypothesis, Aquatic Ape Hypothesis, Markan Priority, Two-Source Hypothesis, Riemann Hypothesis, Gaia Hypothesis, Recent African Origin of Modern Humans, Ancient Astronauts, Augustinian Hypothesis, Proto-Ionians, Odysseus Unbound, Rudolf Falb, Variability Hypothesis, Biophilia Hypothesis, Zoo Hypothesis, Phantom Time Hypothesis, Farrer Hypothesis, Ex-Cubs Factor, Wiseman Hypothesis, Landscape Zodiac, Northwest Germanic, Griesbach Hypothesis, Medea Hypothesis, Cooperative Eye Hypothesis, Ifood, Metasystem Transition, Endurance Running Hypothesis, Carcinisation, Lonely Runner Conjecture, Glasgow Chronology, Broad Spectrum Revolution, Ingvaeonic, Ad Hoc Hypothesis, Compression of Morbidity, Variable Mass Hypothesis. Source: Wikipedia. Pages: 243. 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: In mathematics, the Riemann hypothesis, proposed by Bernhard Riemann (1859), is a conjecture about the distribution of the zeros of the Riemann zeta-function which states that all non-trivial zeros of the Riemann zeta function have real part 1/2. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields. The Riemann hypothesis implies results about the distribution of prime numbers that are in some ways as good as possible. Along with suitable generalizations, it is considered by some mathematicians to be the most important unresolved problem in pure mathematics (Bombieri 2000). The Riemann zeta-function (s) is defined for all complex numbers s 1. It has zeros at the negative even integers (i.e. at s = 2, 4, 6, ...). These are called the trivial zeros. The Riemann hypothesis is concerned with the non-trivial zeros, and states that: The real part of any non-trivial zero of the ...More: http://booksllc.net/?id=19344125 ... Read more

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Chapters: Goldbach's Conjecture, Continuum Hypothesis, Consistency, Diophantine Set, Hilbert's Third Problem, Hilbert's Tenth Problem, Hilbert's Second Problem, Hilbert's Fifth Problem, Hilbert's First Problem, Riemann Hypothesis, Kepler Conjecture, Hilbert's Program, Hilbert's Sixteenth Problem, Hilbert's Twenty-First Problem, Hilbert's Twelfth Problem, Hilbert's Fourteenth Problem, Hilbert's Seventeenth Problem, Hilbert's Sixth Problem, Hilbert's Thirteenth Problem, Hilbert's Eighteenth Problem, Hilbert Number, Hilbert's Seventh Problem, Hilbert's Ninth Problem, Hilbert's Nineteenth Problem, Zetagrid, Hilbert's Eleventh Problem, Hilbert's Fourth Problem, Hilbert's Eighth Problem, Hilbert's Fifteenth Problem, Hilbert's Twentieth Problem, Hilbert's Twenty-Second Problem, Hilbert's Twenty-Third Problem. Source: Wikipedia. Pages: 155. 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: In mathematics, the Riemann hypothesis, proposed by Bernhard Riemann (1859), is a conjecture about the distribution of the zeros of the Riemann zeta-function which states that all non-trivial zeros of the Riemann zeta function have real part 1/2. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields. The Riemann hypothesis implies results about the distribution of prime numbers that are in some ways as good as possible. Along with suitable generalizations, it is considered by some mathematicians to be the most important unresolved problem in pure mathematics (Bombieri 2000). The Riemann zeta-function (s) is defined for all complex numbers s 1. It has zeros at the negative even integers (i.e. at s = 2, 4, 6, ...). These are called the trivial zeros. The Riemann hypothesis is concerned with the non-trivial zeros, and states that: The real part of any...More: http://booksllc.net/?id=19344125 ... Read more

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Georg Cantor. Set Theory, Cantor´s Theorem, Transfinite Number, Leopold Kronecker, David Hilbert, ETH Zurich, Continuum Hypothesis, Bijection, Cantor Cube, Cantor Space, Back-and-Forth Method, Cantor Function, Heine?Cantor Theorem, Controversy over Cantor´s Theory, Men of Mathematics, Ludwig Wittgenstein, Richard Dedekind, Article Sources and Contributors, Image Sources, Licenses and Contributors ... 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. Excerpt: In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis, advanced by Georg Cantor in 1877, about the possible sizes of infinite sets. It states: There is no set whose cardinality is strictly between that of the integers and that of the real numbers.Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's twenty-three problems presented in the year 1900. The contributions of Kurt Gödel in 1940 and Paul Cohen in 1963 showed that the hypothesis can neither be disproved nor be proved using the axioms of ZermeloFraenkel set theory, the standard foundation of modern mathematics, provided set theory is consistent. The name of the hypothesis comes from the term the continuum for the real numbers. Two sets are said to have the same cardinality or cardinal number if there exists a bijection (a one-to-one correspondence) between them. Intuitively, for two sets S and T to have the same cardinality means that it is possible to "pair off" elements of S with elements of T in such a fashion that every element of S is paired off with exactly one element of T and vice versa. Hence, the set {banana, apple, pear} has the same cardinality as {yellow, red, green}. With infinite sets such as the set of integers or rational numbers, this becomes more complicated to demonstrate. The rational numbers seemingly form a counterexample to the continuum hypothesis: the rationals form a proper superset of the integers, and a proper subset of the reals, so intuitively, there are more rational numbers than integers, and fewer rational numbers than real numbers. However, this intuitive analysis does not take account of the fact that all three sets are infinite. It turns out the rational numbers can actually be placed in one-to-o... More: http://booksllc.net/?id=5705 ... Read more

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Chapters: Countable set, Continuum hypothesis, Uncountable set, Cardinality, Infinite set, Set-theoretic definition of natural numbers, Dedekind-infinite set, Cofiniteness, Transfinite number, Equinumerosity, Limitation of size, Cocountability, Transfinite arithmetic,. Source: Wikipedia. Pages: 62. 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: In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor. The elements of a countable set can be counted one at a time - although the counting may never finish, every element of the set will eventually be associated with a natural number. Some authors use countable set to mean a set with the same cardinality as the set of natural numbers. The difference between the two definitions is that under the former, finite sets are also considered to be countable, while under the latter definition, they are not considered to be countable. To resolve this ambiguity, the term at most countable is sometimes used for the former notion, and countably infinite for the latter. The term denumerable is also used to mean countably infinite. A set S is called countable if there exists an injective function from S to the natural numbers If f is also surjective, thus making f bijective, then S is called countably infinite. As noted above, this terminology is not universal: some authors define countable not to include finite sets, i.e. they define countable to mean what is here called "countably infinite". There are alternative (equivalent) formulations of the definition in terms of a bijective function or a surjective function as the following theorem shows. A proof of this result can be found in Lang's text...More: http://booksllc.net/?id=6026 ... 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: Continuum Hypothesis, Forcing, Boolean-Valued Model, List of Forcing Notions, Complete Boolean Algebra, Rasiowa-sikorski Lemma, Easton's Theorem, Martin's Maximum, Countable Chain Condition, Generic Filter, Ramified Forcing, Nice Name, Delta Lemma. Excerpt:In mathematical logic , a Boolean-valued model is a generalization of the ordinary Tarskian notion of structure from model theory . In a Boolean-valued model, the truth values of propositions are not limited to "true" and "false", but instead take values in some fixed complete Boolean algebra . Boolean-valued models were introduced by Dana Scott , Robert M. Solovay , and Petr Vop nka in the 1960s in order to help understand Paul Cohen 's method of forcing . They are also related to Heyting algebra semantics in intuitionistic logic . Definition Fix a complete Boolean algebra B and a first-order language L ; the signature of L will consist of a collection of constant symbols, function symbols, and relation symbols. A Boolean-valued model for the language L consists of a universe M , which is a set of elements (or names ), together with interpretations for the symbols. Specifically, the model must assign to each constant symbol of L an element of M , and to each n -ary function symbol f of L and each n -tuple a0,...,a n -1 of elements of M , the model must assign an element of M to the term f (a0,...,a n -1). Interpretation of the atomic formulas of L is more complicated. To each pair a and b of elements of M , the model must assign a truth value || a = b || to the expression a = b ; this truth value is taken from the Boolean algebra B . Similarly, for each n -ary relation symbol R of L and each n -tuple a0,...,a n -1 of elements of M , the model must assign an element of B to be the truth value || R (a0,...,a n -1)||. Inte... ... 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: Countable set, Cantor's diagonal argument, Surreal number, Continuum hypothesis, Hyperreal number, Extended real number line, Uncountable set, Where Mathematics Comes From, Absolute Infinite, Ultrafinitism, Infinitesimal, Infinite monkey theorem, Actual infinity, Non-standard calculus, Real projective line, Cardinality of the continuum, Temporal finitism, Aleph number, Beth number, Line at infinity, Plane at infinity, Point at infinity, Hyperplane at infinity, Infinity plus one, Superreal number, Hyperinteger, Circular points at infinity, Directed infinity,. Source: Wikipedia. Pages: 189. 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: In mathematics, the surreal number system is an arithmetic continuum containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surreals share many properties with the reals, including a total order ≤ and the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an ordered field. In a rigorous set theoretic sense, the surreal numbers are the largest possible ordered field; all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbers, and the hyperreal numbers, are subfields of the surreals. The surreals also contain all transfinite ordinal numbers reachable in the set theory in which they are constructed. The definition and construction of the surreals is due to John Horton Conway. They were introduced in Donald Knuth's 1974 book Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness. This book is a mathematical novelette, and is notable as one of the rare cases where a new mathematical idea was first pres...http://booksllc.net/?id=51432 ... Read more

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This digital document is a journal article from Behaviour Research and Therapy, published by Elsevier in . 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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Employing the autogenous-reactive model of obsessions (Behaviour Research and Therapy 41 (2003) 11-29), this study sought to test a hypothesized continuum where reactive obsessions fall in between autogenous obsessions and worry with respect to several thought characteristics concerning content appraisal, perceived form, and thought triggers. Nonclinical undergraduate students (n=435) were administered an online packet of questionnaires designed to examine the three different types of thoughts. Main data analyses included only those displaying moderate levels of obsessions or worries (n=252). According to the most distressing thought, three different groups were formed and compared: autogenous obsession (n=34), reactive obsession (n=76), and worry (n=142). Results revealed that (a) relative to worry, autogenous obsessions were perceived as more bizarre, more unacceptable, more unrealistic, and less likely to occur; (b) autogenous obsessions were more likely to take the form of impulses, urges, or images, whereas worry was more likely to take the form of doubts, apprehensions, or thoughts; and (c) worry was more characterized by awareness and identifiability of thought triggers, with reactive obsessions through these comparisons falling in between. Moreover, reactive obsessions, relative to autogenous obsessions, were more strongly associated with both severity of worry and use of worrying as a thought control strategy. Our data suggest that the reactive subtype represents more worry-like obsessions compared to the autogenous subtype. ... Read more

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We prove that the Continuum Hypothesis is equivalent to the Axiom of Choice. Thus, the Negation of the Continuum Hypothesis, is equivalent to the Negation of the Axiom of Choice.The Non-Cantorian Axioms impose a Non-Cantorian definition of cardinality, that is different from Cantors cardinality imposed by the Cantorian Axioms.The Non-Cantorian Theory is the Zermelo-Fraenkel Theory with the Negation of the Axiom of Choice, and with the Negation of the Continuum Hypothesis. This Theory has distinct infinities.ISBN 0980128714ISBN-13 9780980128710 ... Read more

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This digital document is an article from Science and Its Times, brought to you by Gale®, a part of Cengage Learning, a world leader in e-research and educational publishing for libraries, schools and businesses.The length of the article is 1650 words.The article is delivered in HTML format and is available in your Amazon.com Digital Locker immediately after purchase.You can view it with any web browser.The histories of science, technology, and mathematics merge with the study of humanities and social science in this interdisciplinary reference work. Essays on people, theories, discoveries, and concepts are combined with overviews, bibliographies of primary documents, and chronological elements to offer students a fascinating way to understand the impact of science on the course of human history and how science affects everyday life. Entries represent people and developments throughout the world, from about 2000 B.C. through the end of the twentieth century. ... Read more