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*3.8 out of 5 stars*

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- Set Theory and the Continuum Hypothesis (Dover Books on Mathematics)
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ByGary L. Hersteinon December 29, 2016

Brilliant analysis that includes Cohen's fundamental re-envisioning of the structure of proof. Besides the basic introduction to set theory, this book assembles Cohen's work on "forcing," a method of interpreting models that redefined the idea of mathematical infinity. The Kindle edition is well executed. Anyone with even a casual interest in formal logic needs to have this book on their (virtual) book shelf.

4 people found this helpful

ByEduardo Regoon September 4, 2015

The mathematics content deserves a 6 star rating. It was a good surprise to find in this classic of Cohen such a fine exposition, clear and magnificently structured... The poor rating of 2 stars I'm giving it's due to the appalling editing and formating: Amazon should actually be sued for selling a book on mathematics like this. Displayed formulas come in a small type and light gray almost impossible to read, since it doesn't respond to the enlargement option (I had to use a magnifying lens) and the mistakes are too many, with formulas placed at wrong places, maths symbols transformed into strange combination of characters in the main text (as Pi_i in a displayed list of formulas, which became a few lines down &8;i as if the editing had been done by someone who doesn't know any mathematics... Since the exposition is so good it was always possible to spot the errors and made for them, but it will not be the case in other maths books, where unsolvable confusion may be given to the reader...

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ByGary L. Hersteinon December 29, 2016

Brilliant analysis that includes Cohen's fundamental re-envisioning of the structure of proof. Besides the basic introduction to set theory, this book assembles Cohen's work on "forcing," a method of interpreting models that redefined the idea of mathematical infinity. The Kindle edition is well executed. Anyone with even a casual interest in formal logic needs to have this book on their (virtual) book shelf.

ByDr. James V. Blowerson February 4, 2013

This has been one of my favorite books over the years. Copies were hard to get. There was one at a library near my former workplace, which, unintuitively enough, was an Army post; I am not sure how c = aleph_1 applies to Army logistics. I checked it out and read sections of this book. When I retired in 2005, I still had the book and renewed it by email. Eventually the Army wanted it back, so I mailed it back, and so I no longer had the book. I am glad to see, then, that Dover reprinted the book, and so once again I have it. It is a classic work. In it Cohen presents his proof that c = aleph_1 cannot be proved in normal (ZFC) set theory, by introducing a technique he calls "forcing". He also explains many other parts of set theory as well.

1 helpful vote

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ByJoe Shipmanon July 4, 2003

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.

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.

88 helpful votes

89 helpful votes

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ByIsao Nakanoon December 26, 2010

Easiliy readable and very profound in content. Not only a must reading for researchers in set theory but also the best introduction in mathematical logic.

1 helpful vote

2 helpful votes

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ByL. Hotchkisson October 28, 2012

We used to think that if we could prove something existed with symbols, words, or diagrams, that it must exist. However, Cohen and Godel challenge this notion, in a similar way to Whitehead and Russel regarding the completeness of logic and math. By using logic, language, and symbolic logic, Cohen shows that our feeble attempts to prove the continuum exists will have to depend on scientific experiment and contact. This is similar to the problems encountered in modern cosmology. Without a star ship to allow us to boldly go, we cannot verify the existence of many of the images we see through our telescopes, and detectors. Observation from afar can only be supported by assumptions that allow us to educatedly guess whats our there, and how far away it is. But we really need to get out in space, find out how warped space is by gravity, and so on. . .

This is a challenge to us all. We can no more prove the continuum exists than we can prove that god or the devil exists.

This is a challenge to us all. We can no more prove the continuum exists than we can prove that god or the devil exists.

1 helpful vote

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ByHERNAN WEINTRAUBon June 7, 2010

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

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

5 helpful votes

6 helpful votes

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ByVincent Poirieron August 13, 2012

As a work of science, "Set Theory and the Continuum Hypothesis" stands on a par with Darwin's "On the Origin of Species". First, like Darwin's book, Cohen's work is a profound contribution to its field; second it is also accessible to any educated and interested reader, although with some effort.

This edition is a reproduction of the first edition. You might be shocked by the type-this is a plain, typewritten document with no illustrations (I find it charming)-but Paul Cohen's crystal clear prose makes the book eminently readable.

=WHAT YOU NEED=

This is a graduate level book but you don't need to be a graduate student in mathematics to understand it. You do need a laymen's interest in mathematics; for instance you should enjoy reading Euclid, Ian Stewart, Douglas Hoftstadter, Martin Gardner. If you've enjoyed Douglas Hofstadter's "Gödel, Escher, and Bach" then there is no reason you can't understand this book.

=WHAT IT DELIVERS=

First, Cohen gives a barebones but complete introduction to formal logic and logical notation.

Then he describes formal set theory, known as Zemerlo Frankel set theory, the foundation of all mathematics as it stands today.

Having spent half the book on the necessary background, Cohen arrives to his main topic, the Continuum Hypothesis and whether it is true or false.

=WHAT COHEN SAYS=

ST&CH proves that a long standing problem in mathematics (the Continuum Hypothesis) has no solution. What does this mean?

Most mathematicians believe in a scaled down version of Hilbert's Programme. Hilbert hoped that all of mathematics followed from a small collection of definitions and axioms, much like all of geometry was once believed to follow from Euclid's five axioms. Formal set theory, as defined by Zemerlo and Frankel, seemed to provide all the axioms needed for this task. However Kurt Gödel proved that the programme is impossible to realize: any formal system will have propositions that are possible to state but impossible to prove. In other words, no set of axioms can completely define all of mathematics.

Paul Cohen proved that the Continuum Hypothesis is one such statement. But what is this hypothesis? It's about cardinal numbers. A cardinal is a property of a set; specifically it says how many elements there are in a set. For example, the cardinal for the set of all positive odd integers smaller than ten is five because the set {1, 3, 5, 7, 9} has five elements in it.

What about infinite sets? The simplest infinite set we know is the set of Natural Numbers, call it N, and N={1,2,3, ...} . N has an infinite number of elements. What about if we add zero as an element and call the new set N*? Do we get a bigger set? In one way, N* is "bigger" than N because it has all the elements N has but it also has an extra element, "0". But that's not the right way to think about big or small when we talking about sets. We want to know if the cardinal of N* is bigger than the cardinal of N. It isn't.

It's easy to see this. Let's create a new set made up of all the possible ways of writing words with the letter "a" and call this set A. Well, obviously A={a, aa, aaa, aaaa, aaaa, ....}. Now it's obvious that A does not contain N or N*, and vice versa. But can we say A is smaller than or bigger than or the same size as N or N*? Yes we can. Let's start with the natural numbers N. We can say 1 is the first element of N, that 2 is the second element of N, that 3 is the third, and so on. Likewise, we can say a is the first element of A, aa is the second element of A, aaa, is the third, and so on.

Now, bear with me here. We can also say that 0 is the first element of N*, 1 is the second element of N*, 2 is the third element of N*, and so on. So A, N, and N* seem to all have an infinite number of elements that can all be listed, or put in a one-to-one correspondence with each other. They are of the same size, they have the same cardinal, and we call that cardinal number Aleph Null (א is a Hebrew letter). We also say that sets with cardinal Aleph Null are countable, because we can count all their elements one after the other.

The set of positive and negative whole numbers, Z, is also countable. We think of Z={..., -2, -1, 0, 1, 2, 3,...} but we can also write Z={0, 1,-1, 2, -2, 3, -3,...} and it's now easy to see that Z is countable. Surprisingly, the set of all rational numbers (Q, the set of all fractions and whole numbers) is also countable. A rational number is a ratio of two whole numbers, a/b where b is never 0. We can certainly list all pairs of whole numbers in a set called P={(0,1), (1,1), (0,-1), (1,-1),(-1,1), (-1, -1), (0,2),(1,2),(2,2),(-1,2),...}. Since many of these pairs reduce to the same thing, for example (1,2) and (2,4) are both the same as 0.5, Q is a subset of P. So if P is countable, Q is countable.

But what about other numbers? The set of real numbers, called R, is the set of all numbers that can be represented by a point on a line. All the rational numbers (Q) can be represented by a point on a line, but there are many numbers on the line that are not rational. The square root of two or Pi are two famous examples. Are all the numbers on the line countable? It turns out that they are NOT countable. No matter how you list them, you will always find a number that cannot fit anywhere in the list you made. The set R is not countable, so we say it is uncountable.

It is in this sense that we say the set of Real Numbers is bigger than the set of Natural Numbers. We say that the cardinal of R is Aleph One. The Continuum Hypothesis states that there are no cardinals between Aleph Null and Aleph One; that there is no such thing as a set that is bigger than the Natural Numbers but smaller than the Real Numbers.

We owe the above discoveries to a nineteenth century German mathematician named Georg Cantor. He first stated the Continuum Hypothesis and he spent years trying unsuccessfully to prove it. In the 1930s, Kurt Gödel proved that if you assumed that the Hypothesis was true, you did not contradict formal set theory.

In 1964 Paul Cohen proved that if you assumed the Hypothesis was false, you did not contradict formal set theory either. And so he shows that in the context of set theory the Continuum Hypothesis is unprovable.

What is now the way forward? Cohen thinks that one day we will feel the Hypothesis is obviously false. (He underlines the word "obviously".) This means that set theory will have to be perfected, perhaps by adding a single simple axiom that is "obvious" and that results, as a consequence, in a proof that the Hypothesis is false. But with the same humility we find in Darwin, he leaves the problem for future generations to solve.

Vincent Poirier, Montreal

This edition is a reproduction of the first edition. You might be shocked by the type-this is a plain, typewritten document with no illustrations (I find it charming)-but Paul Cohen's crystal clear prose makes the book eminently readable.

=WHAT YOU NEED=

This is a graduate level book but you don't need to be a graduate student in mathematics to understand it. You do need a laymen's interest in mathematics; for instance you should enjoy reading Euclid, Ian Stewart, Douglas Hoftstadter, Martin Gardner. If you've enjoyed Douglas Hofstadter's "Gödel, Escher, and Bach" then there is no reason you can't understand this book.

=WHAT IT DELIVERS=

First, Cohen gives a barebones but complete introduction to formal logic and logical notation.

Then he describes formal set theory, known as Zemerlo Frankel set theory, the foundation of all mathematics as it stands today.

Having spent half the book on the necessary background, Cohen arrives to his main topic, the Continuum Hypothesis and whether it is true or false.

=WHAT COHEN SAYS=

ST&CH proves that a long standing problem in mathematics (the Continuum Hypothesis) has no solution. What does this mean?

Most mathematicians believe in a scaled down version of Hilbert's Programme. Hilbert hoped that all of mathematics followed from a small collection of definitions and axioms, much like all of geometry was once believed to follow from Euclid's five axioms. Formal set theory, as defined by Zemerlo and Frankel, seemed to provide all the axioms needed for this task. However Kurt Gödel proved that the programme is impossible to realize: any formal system will have propositions that are possible to state but impossible to prove. In other words, no set of axioms can completely define all of mathematics.

Paul Cohen proved that the Continuum Hypothesis is one such statement. But what is this hypothesis? It's about cardinal numbers. A cardinal is a property of a set; specifically it says how many elements there are in a set. For example, the cardinal for the set of all positive odd integers smaller than ten is five because the set {1, 3, 5, 7, 9} has five elements in it.

What about infinite sets? The simplest infinite set we know is the set of Natural Numbers, call it N, and N={1,2,3, ...} . N has an infinite number of elements. What about if we add zero as an element and call the new set N*? Do we get a bigger set? In one way, N* is "bigger" than N because it has all the elements N has but it also has an extra element, "0". But that's not the right way to think about big or small when we talking about sets. We want to know if the cardinal of N* is bigger than the cardinal of N. It isn't.

It's easy to see this. Let's create a new set made up of all the possible ways of writing words with the letter "a" and call this set A. Well, obviously A={a, aa, aaa, aaaa, aaaa, ....}. Now it's obvious that A does not contain N or N*, and vice versa. But can we say A is smaller than or bigger than or the same size as N or N*? Yes we can. Let's start with the natural numbers N. We can say 1 is the first element of N, that 2 is the second element of N, that 3 is the third, and so on. Likewise, we can say a is the first element of A, aa is the second element of A, aaa, is the third, and so on.

Now, bear with me here. We can also say that 0 is the first element of N*, 1 is the second element of N*, 2 is the third element of N*, and so on. So A, N, and N* seem to all have an infinite number of elements that can all be listed, or put in a one-to-one correspondence with each other. They are of the same size, they have the same cardinal, and we call that cardinal number Aleph Null (א is a Hebrew letter). We also say that sets with cardinal Aleph Null are countable, because we can count all their elements one after the other.

The set of positive and negative whole numbers, Z, is also countable. We think of Z={..., -2, -1, 0, 1, 2, 3,...} but we can also write Z={0, 1,-1, 2, -2, 3, -3,...} and it's now easy to see that Z is countable. Surprisingly, the set of all rational numbers (Q, the set of all fractions and whole numbers) is also countable. A rational number is a ratio of two whole numbers, a/b where b is never 0. We can certainly list all pairs of whole numbers in a set called P={(0,1), (1,1), (0,-1), (1,-1),(-1,1), (-1, -1), (0,2),(1,2),(2,2),(-1,2),...}. Since many of these pairs reduce to the same thing, for example (1,2) and (2,4) are both the same as 0.5, Q is a subset of P. So if P is countable, Q is countable.

But what about other numbers? The set of real numbers, called R, is the set of all numbers that can be represented by a point on a line. All the rational numbers (Q) can be represented by a point on a line, but there are many numbers on the line that are not rational. The square root of two or Pi are two famous examples. Are all the numbers on the line countable? It turns out that they are NOT countable. No matter how you list them, you will always find a number that cannot fit anywhere in the list you made. The set R is not countable, so we say it is uncountable.

It is in this sense that we say the set of Real Numbers is bigger than the set of Natural Numbers. We say that the cardinal of R is Aleph One. The Continuum Hypothesis states that there are no cardinals between Aleph Null and Aleph One; that there is no such thing as a set that is bigger than the Natural Numbers but smaller than the Real Numbers.

We owe the above discoveries to a nineteenth century German mathematician named Georg Cantor. He first stated the Continuum Hypothesis and he spent years trying unsuccessfully to prove it. In the 1930s, Kurt Gödel proved that if you assumed that the Hypothesis was true, you did not contradict formal set theory.

In 1964 Paul Cohen proved that if you assumed the Hypothesis was false, you did not contradict formal set theory either. And so he shows that in the context of set theory the Continuum Hypothesis is unprovable.

What is now the way forward? Cohen thinks that one day we will feel the Hypothesis is obviously false. (He underlines the word "obviously".) This means that set theory will have to be perfected, perhaps by adding a single simple axiom that is "obvious" and that results, as a consequence, in a proof that the Hypothesis is false. But with the same humility we find in Darwin, he leaves the problem for future generations to solve.

Vincent Poirier, Montreal

30 helpful votes

31 helpful votes

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ByJ. Holton November 5, 2012

Paul Cohen invented forcing as a method for proving the independence of the Continuum Hypothesis from the axioms of set theory, but his approach to the independence proof--which is presented in the last chapter of this book--has been superseded by other approaches (e.g. boolean-valued models) that are easier to follow and more mathematically enlightening. Still, I would highly recommended this little book for two reasons. First, its rapid traversal of first-order logic--completeness, compactness, etc.--is wonderfully lucid and concise: a real mathematician's treatment, blessedly free of the logician's "hair." The same is true of his treatment of axiomatic set theory. His proof, for example, of the Schroeder-Bernstein theorem (if there is an injection from A into B and from B into A, then there is a bijection between A and B) is the briefest and most elegant (and most intuitive) I've ever seen--about three lines, as compared with the usual three-quarters of a page. The second thing I enjoyed about this book is Cohen's many fascinating obiter dicta--for example, when he conjectures near the end that the continuum, being "an incredibly rich set given to us by a bold new axiom [the power-set axiom]," may well have a cardinality that outstrips any aleph obtainable from the replacement axiom. Cohen is as fresh and intuitive as he is intrepid.

The volume has very interesting prefatory memoir by Cohen in which he recounts how he was attracted to the Continuum Problem and his meetings with Kurt Gödel after he (Cohen) obtained his independence proof. There is also a nice introduction by the great logician Martin Davis.

The volume has very interesting prefatory memoir by Cohen in which he recounts how he was attracted to the Continuum Problem and his meetings with Kurt Gödel after he (Cohen) obtained his independence proof. There is also a nice introduction by the great logician Martin Davis.

6 helpful votes

7 helpful votes

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ByEduardo Regoon September 4, 2015

The mathematics content deserves a 6 star rating. It was a good surprise to find in this classic of Cohen such a fine exposition, clear and magnificently structured... The poor rating of 2 stars I'm giving it's due to the appalling editing and formating: Amazon should actually be sued for selling a book on mathematics like this. Displayed formulas come in a small type and light gray almost impossible to read, since it doesn't respond to the enlargement option (I had to use a magnifying lens) and the mistakes are too many, with formulas placed at wrong places, maths symbols transformed into strange combination of characters in the main text (as Pi_i in a displayed list of formulas, which became a few lines down &8;i as if the editing had been done by someone who doesn't know any mathematics... Since the exposition is so good it was always possible to spot the errors and made for them, but it will not be the case in other maths books, where unsolvable confusion may be given to the reader...

4 helpful votes

5 helpful votes

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ByAmazon Customeron April 27, 2007

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.

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.

51 helpful votes

52 helpful votes

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