On Formally Undecidable Propositions of Principia Mathematica and Related Systems

Book Description

First English translation of revolutionary paper (1931) that established that even in elementary parts of arithmetic, there are propositions which cannot be proved or disproved within the system. It is thus uncertain that the basic axioms of arithmetic will not give rise to contradictions. Introduction by R. B. Braithwaite.

Customer Reviews:

Mathematical Rationalism has limits.......2007-03-17

It is very hard to find faults in what may be the most famous proof of the 20th century.
For those not familiar with the Russell-Whitehead Principia Mathematica notation
this is a very hard book. I had the benefit of the Kac-Ulam explanation.
I did find what might be problems with this proof.
1) One is the reliance on number theory proofs about prime numbers that are assumed true
in the Gödelization of the primes coding of the mathematical axioms.
2) The second is the assumption that the axioms statements represent the minimal
representation of such a system of axioms.
Both are slim if none chances, but ones the Gödel doesn't consider.
Information theory was after this time where we discovered that a system of symbols can indeed at times be more efficiently coded.
The best example of this seems to be Gray code compared to ordinary binary number code ( a number theory code
like Gödel's prime code) where less turns out to be more in information terms.
The theory of primes suffers from the new doctrine of strings that says
that infinite scales don't exist in the "real" world: that a maximum and a minimum
of measure are fixed parts of our reality. This kind of assumption can't be "proved"
but is an axiom of a system of a mathematical sort and is counter to the Euclidean proof of an infinite number of primes.
Primes already discovered by use of computers are much bigger than the numbers of ordinary physics, but
we are already reaching the Turing "stopping" problem in finding new ones.
Some people equate in algorithmic information theory and number theory
the stopping problem with Infinity. That point of view of people like G. J. Chaitin
is itself an unproved assumption. So the metamathematics used in the proof itself may be unprovable propositions.
If so, then the proof based on such propositions can't itself be true.
This argument in no way takes away from the greatness of Gödel and his unique genius
as shown by this line of reasoning.

The following is a dissenting view.......2006-10-25

As indicated in two other reviews of mine here, my comprehension of Goedel's work is opposite to the general one. My marking three stars regardless for this book is motivated by his extensive influence, but also by his fair admission later in life that his thesis could amount to hocus-pocus.

Indeed, I see it as one of the prominent mistakes in logical history, and I shall endeavor to explain as best I can. It should suffice to consider his Section 1, an outline of his proposed proof.

Although that section is brief, it already foreshadows an oppressingly complex logical symbolism for statements that in my view can be made much clearer using ordinary language. The symbolism, to be sure, is intended to establish a formal language, whose meaning is to be decided separately. This will be seen one of the problems.

For now, let me give the principal statement Goedel contended to be true but undecidable (neither provable nor disprovable):

"This statement is unprovable."

He symbolized it (p.40) as: "~Bew[R(n);n]". Font limitations made me slightly change it; the tilde "~" means "not", "Bew" is a German abbreviation for "provable", and within brackets "R(n)" says "Statement n" and "n" stands for the full statement.

Goedel proceeds: "...supposing...~Bew[R(n);n] were provable, it would also be correct; but that means...that...~Bew[R(n);n] would hold good, in contradiction to our initial assumption. If, on the contrary, the negation of ~Bew[R(n);n] were provable, then [its provability] would hold good. ~Bew[R(n);n] would thus be provable [in contradiction to the unprovability it states], which again is impossible." (I corrected some errors within brackets.)

So since both ~Bew[R(n);n] and its negation are unprovable, it is undecidable, and Goedel continues (p.41): "...it follows at once that ~Bew[R(n);n] is correct, since...certainly unprovable (because undecidable). So the proposition which is undecidable in the system...turns out to be decided by metamathematical considerations."

"Metamathematical", in excusing the contradiction, designates the above formal system void of assigned meaning, whereas the statement discussed is to have meaning. Not quite a lucid argument. Overlooked, furthermore, is a contradiction using the same reasoning as in the preceding.

Coupled with the preceding finding that ~Bew[R(n);n] CANNOT be proved unprovable (for if so proved, it would be contradicted), can in contradiction be that it CAN be proved unprovable. For if it were instead provable, it would again be contradicted. The statement in question thus becomes a paradox, rather than true, similar to paradoxes like the "liar", mentioned by Goedel (p.40).

He strangely adds to it the footnote: "Every epistemological [paradox] can likewise be used for a similar undecidability proof." The "liar", however, is, like all paradoxes, not a true statement, as required, but one harboring a contradiction. (I deal in my book with, and offer solutions to, paradoxes more fully, including Goedel's resulting one, without naming him.)

There occurs, further, another huge blunder in the alleged proof. The undecidability is said to apply to some of mathematics; in the above formula, ~Bew[R(n);n], the "n" refers to a number, with this justification by Goedel (p.38): "For metamathematical purposes it is naturally immaterial what objects are taken as basic signs, and we propose to use natural numbers for them." Adding (p.39): "Metamathematical concepts and propositions thereby become concepts and propositions concerning natural numbers..."

How so? In one breath he proposes using natural numbers as immaterial signs, and in the next breath the material concerns natural numbers!

The fallaciousness can indeed be made clear by considering our statement, ~Bew[R(n);n], interpreted as "This statement is unprovable." As noted, in ~Bew[R(n);n] the "n", now a number, is to name the whole statement, inside which it is also used in "Statement n..." But whether or not the statement is named by a number, the point is that the name must refer to the intended content of the statement to correspondingly function, not to the usual number possibly represented. Therefore the statement, or anything else similarly used, has nothing to do with numbers, or mathematics generally.

Gödel's proof of the inadequacy of formalism.......2006-10-16

Gödel proves that a formal system containing arithmetic must be incomplete (i.e. incapable of proving all true statements). The proof consists in creating a statement that says "this statement cannot be proved", for then it follows that either this this statement can be proved and we have proved something false, or it cannot be proved but it is still true. In either case our formal system is flawed. This is in a way an instance of the liar paradox, which was of course well know long before, but no-one had expected it to materialise inside a seemingly sensible formal system. Gödel shows that it does by means of his arithmetisation trick that enables the system to speak about itself. All symbols in the system's alphabet is given a unique number. Then all formulas in the system is assigned the following number: the product of all the factors (n:th prime)^(n:th symbol in the formula). By unique prime factorisation one can recreate the formula from its number. Sequences of formulas---proofs in particular---can be coded by the same method. We can now express the relation "x is a proof of y" inside the formal system. This relation takes two arguments: x*, the Gödel number for the sequence of formulas x, and y*, the Gödel number for the formula y. Inside the formal system it is a perfectly well defined and finite problem to decide whether x is a proof of y, as is quite plausible, although Gödel has to work hard with his recursion theory to prove this strictly. Now that we can express "x is a proof of y" we can also express "x is a proof of y(z)", i.e. a relation that takes three arguments: x*, y*, z*, the Gödel numbers for a sequence x of formulas, a formula y with a free variable, and a formula z. Thus we can also express "there exists no x such that x is a proof of y(z)". In particular, we can send in y* for z, and the statement becomes: "there exists no x such that x is a proof of y(y*)". This expression has one free variable, y. Call it F(y). F(y) is a formula in our formal system, so it has a Gödel number, say F*. Now we can formulate the statement "this statement cannot be proved" inside our formal system as follows: "F(F*)"="there exists no x such that x is a proof of F(F*)"="F(F*) cannot be proved". So if our formal system is consistent (i.e. does not prove false things) then we must accept that it cannot prove F(F*), but then F(F*) is true, so our formal system is incomplete.

One of the Best Books You Should Never Read.......2005-07-24

Godel's incompleteness theorem's are without a doubt genious. However, this day in age, no logician actually reads Godel's original work unless they are only interested in the historical aspect of it. Godel himself is not a very good writer. If you want to study Godel's incompleteness theorems there are other books out there that prove his theorems in a much more refined, shorter, and easier fasion.

Unbelievable theorem.......2004-08-04

Reading through the reviews of self-proclaimed math geniuses (see some of the below unhelpful reviews for examples) is hardly edifying, so I feel compelled to lend a hand. Here are a few comments about this publication:

First, the introduction does a poor job in explicating the theory. I suppose it gives you the basic idea, but this is hardly the first account of the theory one should read. Brathwaite does not connect all of the dots, and it will take a long time to figure out how the proof works from his intro, if you can do it all. (And that's not a challenge or insult; it simply isn't that well written.)

Second, forget about wading through Godel's proof on your own. The reviewer who claimed to do so with two years of algebra and a really good dictionary is simply lying. You do not wade through difficult theorems in mathematical logic without the appropriate tools. And the appropriate tools include having done similar but simpler proofs on your own and having a solid background in mathematical logic. Without this background, it doesn't matter whether you have the ability to be a mathematics professor at Princeton or place top five in the Putnam - you simply will not understand the proof in a rigorous manner. By all means, take a look at it to get a general feel for what's going on, but if you want a semi-technical account read Smullyan's "Godel's Incompleteness Theorems."

Third, as one reviewer pointed out, there are multiple errors in this printing of the proof. This makes what was a tall task virtually impossible.

So what did Godel do that was so interesting?
He proved that there were certain arithmetical statements about whole numbers that were not provable but true. (This was important because it shattered the widely held belief that if you stated a problem in mathematics clearly enough you would be able to determine whether it was true or false. Godel showed this isn't always the case. As an aside, simpler mathematical systems have been shown complete; that is to say, they can answer any well formed question.)
So, how can something be true but unprovable?
The sentence Godel constructed said this, more or less: I am not provable. This statement, if true, is not provable. If it is provable it's false, and correct systems (systems that do not prove false statements) cannot prove false statements. Therefore, it must not be provable. But then it's saying something true, and thus it's true but unprovable. Now, I'm simplifying and being sloppy, and you need to know about the difference between mathematical statements and metamathematical statements, but in a nutshell that's the thrust of his first theorem.

The other interesting aspect of his proof is that he constructed a statement that referred to itself indirectly. Russell, in Principia Mathematica - the work that contains the arithmetical system that served as the model for the arithmetical system in Godel's proof - created a "Theory of Types" which did not allow statements to mention themselves. But the sentence "I am not provable" references itself so it would seem that I've erred. But in fact I haven't; I just didn't fully explain how that sentence worked. (I know you were worried, if for just an instant.) Where was I . . . Godel created a sentence which referred to itself indirectly. The sentenced said, "Sentences with such and such characteristics are unprovable." It so happened that a sentence with such characteristics was itself. Thus, it referred to itself, but only indirectly and not in violation of the "Theory of Types."

All of my blathering, I hope, has impressed on you . . .
1) That this proof is worth understanding.
2) That you shouldn't believe anyone who tells you they worked through and understood the proof without having a signficant background in mathematical logic and the history of the proof. If you don't understand certain basic features of Principia Mathematica you're not going to grasp fully his proof.
3) That you should get an introductory account. Nagle's "Godel's Proof" is excellent and easy to understand. Smullyan's "Godel's Incompleteness Theorems" is more difficult, but not impossible and amounts to what would serve as the textbook of a solid mathematical logic course or two at an elite university.
4) That you shouldn't buy this work if you're hoping to work through his proof, unless of course you have the requisite training. Brain power is not enough.

From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931 (Source Books in the History of the Sciences)

Average customer rating:

A classic
Within the reach of determined general readers
Oops
Just a comment.
Essential reference in the history of logic and computing

From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931 (Source Books in the History of the Sciences)
Jean van Heijenoort
Manufacturer: Harvard University Press
ProductGroup: Book
Binding: Paperback

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ASIN: 0674324498

Book Description

The fundamental texts of the great classical period in modern logic, some of them never before available in English translation, are here gathered together for the first time. Modern logic, heralded by Leibniz, may be said to have been initiated by Boole, De Morgan, and Jevons, but it was the publication in 1879 of Gottlob Frege's Begriffsschrift that opened a great epoch in the history of logic by presenting, in full-fledged form, the propositional calculus and quantification theory.

Frege's book, translated in its entirety, begins the present volume. The emergence of two new fields, set theory and foundations of mathematics, on the borders of logic, mathematics, and philosophy, is depicted by the texts that follow. Peano and Dedekind illustrate the trend that led to Principia Mathematica. Burali-Forti, Cantor, Russell, Richard, and König mark the appearance of the modern paradoxes. Hilbert, Russell, and Zermelo show various ways of overcoming these paradoxes and initiate, respectively, proof theory, the theory of types, and axiomatic set theory. Skolem generalizes Löwenheim's theorem, and he and Fraenkel amend Zermelo's axiomatization of set theory, while von Neumann offers a somewhat different system. The controversy between Hubert and Brouwer during the twenties is presented in papers of theirs and in others by Weyl, Bernays, Ackermann, and Kolmogorov. The volume concludes with papers by Herbrand and by Gödel, including the latter's famous incompleteness paper.

Of the forty-five contributions here collected all but five are presented in extenso. Those not originally written in English have been translated with exemplary care and exactness; the translators are themselves mathematical logicians as well as skilled interpreters of sometimes obscure texts. Each paper is introduced by a note that sets it in perspective, explains its importance, and points out difficulties in interpretation. Editorial comments and footnotes are interpolated where needed, and an extensive bibliography is included.

Customer Reviews:

A classic.......2007-01-10

This book contains translations of original articles from this period. In one case, Herbrand's theorem, there are extensive notes to repair a mistake; but most are simply presented as is, with short introductions that give some historical context. It is really wonderful to see the ideas develop. Fortunately, this book has recently been reprinted. Library copies are falling apart.

Within the reach of determined general readers.......2006-01-20

This excellent collection has introductions which help immensely. With only a math major from the 50's and no advanced degree I was still able to develop my own fairly rigorous single page synopsis of Godel's theorems.

Oops.......2003-03-10

"Philosophical and Mathematical Correspondence" was published 13 years after Heijenoort's.

Just a comment........2003-03-10

In response to Jay Miller's question below there is a book titled "Philosophical and Mathematical Correspondence. Gottlob Frege" that has 21 letters between Russell and Frege over a period of 10 years beginning with Russell's observation of his famous paradox in 1902. This wonderful collection of correspondence was published 20 years before "From Frege to Godel" and includes letters from many of the same mathematicians and logicians.

Essential reference in the history of logic and computing.......2002-12-13

The second part of my review title may shock some, but the excellent collection of papers that Van Heijenoort has edited (and in many cases translated!) is also an excellent reference in the history of computing. Everyone appreciates that mathematical logic gave rise to computer science; the papers in this collection from Hilbert, Herbrand, Gödel, and others will show why.

If your interest is instead the history of logic, all the classics in the range specified by the work's title are here, complete with their own ideosyncratic notation. van Heijenoort's wonderful introductions to each piece will interelate the works, provide references to other literature and situate everything in a wonderful intellectual climate.

Be warned, however, that the foundational papers in this still growing field continue for another 15 years or so; these are reprinted in Davis' (alas, out of print) anthology _The Undecidable_.

Book Description

This is a book about the philosophy of time, and in particular the philosophy of the great logician Kurt Godel (1906-1978). It evaluates Godel's attempt to show that Einstein has not so much explained time as explained it away. Unlike recent more technical studies, it focuses on the reality of time. The book explores Godel's conception of time, existence, and truth with special reference to Plato, Aristotle, Kant, and Frege. In the light of this investigation an attempt is made to shed light on such issues as the precise sense in which Godel believed in the possibility of time travel, the relationship of the reality of time to the objectivity of temporal becoming, and the significance of time for human existence.This is a book about the philosophy of time, and in particular the philosophy of the great logician Kurt Godel (1906-1978). It evaluates Godel's attempt to show that Einstein has not so much explained time as explained it away. Unlike recent more technical studies, it focuses on the reality of time. The book explores Godel's conception of time, existence, and truth with special reference to Plato, Aristotle, Kant, and Frege. In the light of this investigation an attempt is made to shed light on such issues as the precise sense in which Godel believed in the possibility of time travel, the relationship of the reality of time to the objectivity of temporal becoming, and the significance of time for human existence.

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Wang Exposes Godel's Great Predictions.
Since no one else has reviewed this I will.

Reflections on Kurt Gödel
Hao Wang
Manufacturer: The MIT Press
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ASIN: 0262730871

Book Description

Kurt Gödel was indisputably one of the greatest thinkers of our time, and in this first extended treatment of his life and work, Hao Wang, who was in close contact with Gödel in his last years, brings out the full subtlety of Gödel's ideas and their connection with grand themes in the history of mathematics and philosophy. The subjects he covers include the completeness of elementary logic, the limits of formalization, the problem of evidence, the concept of set, the philosophy of mathematics, time and relativity theory, metaphysics and religion, as well as general ideas on philosophy as a worldview.

Hao Wang is Professor of Logic at the Rockefeller University.

Customer Reviews:

Wang Exposes Godel's Great Predictions........2005-04-12

On Pages 1 and 2, Wang tells us that Godel, the master of the incomplete, suggests the possibility of philosophy as an exact theory emerging within the next hundred years or even sooner. There will be, he believes, scientific disproofs of what he calls' mechanism in biology' and of the proposition that 'there is no mind separate from matter'; moreover he thinks it practically certain that the 'physical laws, in their observable consequences, have a finite limit of precision. In his conversations, he recommends the important project of finding what might be called a 'rational religion.'

I conclude that exact philosophy already exists because theological statements are being proven, even though the ultimate truth will always be incomplate. This prediction means that the scientific method cannot be used to prove worlds, which is a box in which we live. Thus, universe cannot be measured without measure standards. So the universe is relativistic and can never be known exactly. I also agree with Godel that mechanisms will never be found in living things. This is why US medical care is so bad. I agree with Godel that minds will never be without bodies because only organizations exist in Nature. I also agree with Godel that a rational religion is coming because theological statements are being proven.

Since no one else has reviewed this I will........2004-03-03

Wang has been an important source in compiling information on Godel and bringing it to public attention. This volume contains a variety of material about Godel- biographical facts, personal recollections, chronologies, Godel's philosophical ideas, the impact and historical setting of his mathematical work, his relationship with Einstein, comparisons to other prominent intellectuals, and more. It assumes a basic understanding of Godel's theorems. The bulk of the book is a presentation of some of Godel's (largely unpublished) philosophical activity. There is also quite a bit on Wang's own views as he contrasts them with Godel's. Some of these sections require more background in philosophy than most students of mathematics possess (myself included).

Wang supplies lots of interesting historical and biographical material as well. The 75 page chronology of Godel's life and work is very informative. Contains 11 photographs of Godel and company. The book ends with some useful commentary on selected publications of Godel. If you're looking just for a biography get Dawson's excellent book, but anyone seriously interested in Godel will want this as well.

Average customer rating:

Un understandable overview of Godel and his completeness theorem
An abridged version of Hofstadter's book.
Biography: no -- Look at his great theorm: YES!
Not the real Gödel ?
Not really a biography, but very good nonetheless

Gödel: A Life of Logic
John L. Casti , J. L. Casti , and Werner DePauli
Manufacturer: Perseus Books
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ASIN: 0738205184
Release Date: 2001-09-18

Book Description

"A compelling biography of the eccentric genius."- Discover.

Kurt Gödel was an intellectual giant. His Incompleteness Theorem turned not only mathematics but also the whole world of science and philosophy on its head. Shattering hopes that logic would, in the end, allow us a complete understanding of the universe, Gödel's theorem also raised many provocative questions: What are the limits of rational thought? Can we ever fully understand the machines we build? Or the inner workings of our own minds? How should mathematicians proceed in the absence of complete certainty about their results? Equally legendary were Gödel's eccentricities, his close friendship with Albert Einstein, and his paranoid fear of germs that eventually led to his death from self-starvation. Now, in the first book for a general audience on this strange and brilliant thinker, John Casti and Werner DePauli bring the legend to life.

Customer Reviews:

Un understandable overview of Godel and his completeness theorem.......2006-09-14

The main result of Godel's Completeness Theorem is that in arithmetic, there are true statements that can never be proven to be true in the system of arithmetic. Using this as a base system, this means that in any system equal to or greater than arithmetic in complexity, there will be true statements that cannot be proven to be true in the system. This result has been used by many people to argue for or against many things.
I have seen it used to argue for the existence of God.

"According to Godel's theorem, there are things that are true that cannot be proven to be true within the system of human thought. God is one such thing, therefore God exists."

I have seen it used to argue against the possibility of artificial intelligence (AI).

"According to Godel's theorem, there are things that are true that cannot be proven to be true within the system of programmable human thought. Humans take advantage of these unprovable truths, which makes intelligence. Since this advantage can never be programmed, AI is impossible."

I have suggested on more than one occasion that the people making these arguments need to spend more time studying both logic and what Gödel really concluded. For example, they could read this book.
It presents a brief biography of Kurt Gödel. In his later years he was quite eccentric and reclusive, however in his early years he apparently was quite a ladies man. Certainly Gödel was a genius; Albert Einstein himself openly expressed his admiration for Godel's intelligence. I was pleased to see the authors spend as much time as they did describing Gödel in his earlier years. So many other commentators spend so much time on his social difficulties that his achievements become overshadowed.
A complete explanation of his main results is also expressed in terminology that almost everyone can understand. There are few formulas; simple algebra is all that is needed to understand all of the mathematical symbolism used in the book. If I was teaching a course in popular mathematics, it would have to include Godel's Completeness Theorem and this is the book I would select for that section.

An abridged version of Hofstadter's book. .......2004-12-29

Author shows a great skill in Chapter Two and Three to explain a crux of famous theorem in a very succinct language without using mathematical terms. Also a short biography of Godel's strange life explains: why he died of paranoia; why he hated Austria; why he was suffering a guilt of not producing enough academic result in Princeton.
As the author acknowledges, many metaphors used in this book overlaps with the Douglas Hofstadter's Pulitzer-winning book. However, as many other books for past twenty years, the author presents a theorem in a way that is easily misinterpreted. In p11, he says "Essentially, what Godel showed is that no kind of mathematics is ever going to be comprehensive enough to express fully the everyday notion of truth." And then, the author spends a great deal of pages on AI and computer.
As far as I know, Godel's theorem mentions nothing about "truth in daily life" or computer. Godel's theorem applies only in a strictly circumscribed sphere, i.e., first-order logic. For example, Euclidean geometry is not imcomplete, and the higher-order logic doesn't produce Russel-type paradox. So, what we cannot speak of we must pass over in silence.
Also, author asserts in p71 that Wittgenstein's shift to "sociocultural position" later in his life, but he failed to mention that Wittgenstein did describe his thought about Godel's theorem in his "Remarks on the Foundations of Mathematics".

Biography: no -- Look at his great theorm: YES!.......2003-10-19

I got to look at the book at a bookstore before I bought it so I knew I wasn't getting a biography. This book is a look at his theorem with comments about his life thrown in to put the work into some human context. For a thurough description of the theorem with a gentle human touch this is the book for you. Casti et al. does a great job of making tough ideas readable. If you want to know more about the theorem that turned mathematics on its head this is it. Not perfect (less talk about cake :-) ) but fun, readable, educational, A shame it is out of print.

Not the real Gödel ?.......2003-04-22

Sorry, but this book was somewhat a disappointment for me. The authors for the most part keep personal life and work of Gödel separated, instead of seeing them as a unity. A biography has to be the best of both worlds in my opinion. That's what makes the work of a biographical writer a difficult task. Maybe one of the two authors did the biographical part, the other one the mathematical ? And of course, everything about Gödel is great, brillant and alltogether grand. I am missing a critical view on his lifestyle and his view on music e.g.. Appearently the author of the biographical part was so in awe of Gödel, that he didn't dare to critisize anything about Gödel. Ironic, since Gödel stands for the idea, that you are allowed and even have the obligation to question everything to get to the bottom of the truth of things.
I am still waiting for the real biography of Kurt Gödel.

Not really a biography, but very good nonetheless.......2002-10-24

I would agree with other reviewers who point out that Casti and DePauli's book really doesn't work as a biography. While there are some interesting biographical factoids, they are offered in such a disjoint manner that it is hard to see this book as a good biography of Kurt Godel.

However, as a book that gives an accessible overview of Godel's work, it is very effective. The best parts of the book deal with Godel's Theorem and Turing's Halting Problem. While there are other books out there that do a good job of making both those topics accessible to a wide audience, Casti and DePauli's treatment is worth a read because they also offer some unique insights not (easily) found elsewhere.

But the best part of this book is the second to the last chapter that gives an accessible account of Algorithmic Information Theory (aka 'Kolmogorov Complexity') ... especially Gregory Chaitin's work on the randomness of natural numbers. While Chaitin has also written some accessible works on this topic, Casti and DePauli does a great job of explaining this topic to a wider audience as well as showing the connections between AIT and Godel/Turing. This chapter alone is worth the price of the book.

A very interesting and insightful thing that Casti and DePauli did was to periodically re-define Godel's Theorem in terms of Turing's Halting Problem, Chaitin's work, and from other interesting angles.

The book is not without fault. Besides the rather haphazard biographical details, the chapters dealing with some of Godel's other projects (physics, mysticism, etc.) were rather poorly written. Also, Casti and DePauli did a very bad job with citations/suggestions for further reading. E.g., they often cite to other works, or suggest readers consult other sources for further details, and then do NOT provide those sources in the bibliography. There are some other examples of sloppy editing and writing that would be hard to point out to those who haven't actually read the book.

Book Description

Hao Wang (1921-1995) was one of the few confidants of the great mathematician and logician Kurt Gödel. A Logical Journey is a continuation of Wang's Reflections on Gödel and also elaborates on discussions contained in From Mathematics to Philosophy. A decade in preparation, it contains important and unfamiliar insights into Gödel's views on a wide range of issues, from Platonism and the nature of logic, to minds and machines, the existence of God, and positivism and phenomenology.

The impact of Gödel's theorem on twentieth-century thought is on par with that of Einstein's theory of relativity, Heisenberg's uncertainty principle, or Keynesian economics. These previously unpublished intimate and informal conversations, however, bring to light and amplify Gödel's other major contributions to logic and philosophy. They reveal that there is much more in Gödel's philosophy of mathematics than is commonly believed, and more in his philosophy than his philosophy of mathematics.

Wang writes that "it is even possible that his quite informal and loosely structured conversations with me, which I am freely using in this book, will turn out to be the fullest existing expression of the diverse components of his inadequately articulated general philosophy."

The first two chapters are devoted to Gödel's life and mental development. In the chapters that follow, Wang illustrates the quest for overarching solutions and grand unifications of knowledge and action in Gödel's written speculations on God and an afterlife. He gives the background and a chronological summary of the conversations, considers Gödel's comments on philosophies and philosophers (his support of Husserl's phenomenology and his digressions on Kant and Wittgenstein), and his attempt to demonstrate the superiority of the mind's power over brains and machines. Three chapters are tied together by what Wang perceives to be Gödel's governing ideal of philosophy: an exact theory in which mathematics and Newtonian physics serve as a model for philosophy or metaphysics. Finally, in an epilog Wang sketches his own approach to philosophy in contrast to his interpretation of Gödel's outlook.

Customer Reviews:

Hao Wang, Unsung Hero.......2007-05-16

Wang's presentation of Godel brings the supergenius mathematical logician within the reach of people who are neither logicians nor mathematicians ... at least occasionally. "Godel, Escher and Bach," a previous best-seller effort, didn't manage to do that. I never thought I could or would stay with a book I comprehended so little. It was like digging through a 5 gallon drum of sunflower seeds in search of a cupful of sesame seeds that I could digest and metabolize. But I couldn't stop! Every time I found one of those sesame seeds I could understand and maybe even use to help me understand something else, I got a rush of motivation to keep on reading, in hopes there would be at least one more such sesame seed! The reason was Wang's delivery, based on his very way of being. He is a smart, trained mathematical logician himself who grew up in a contrasting philosophical culture [featuring Chinese nontheistic assumptions] and he managed to become as humble and honest and open minded and open hearted an individual as I have yet encountered in person or on the printed page. His use of self disclosure ... an au currant recommended practice among scientist science writers ... demonstrates a Goldilocks model for others to follow: not too much -- no egotistical tangents, and not too little -- he is remarkably clear about his own assumptons, biases and prejudices. Even if you don't care much about understanding Godel, the book is worth reading to get acquainted with Hao Wang.

The end of books: the pinnacle of knowledge.......2006-04-07

: The ideas expressed in this book are at least 100 years ahead of their time. Godel wasn't just friends with Einstein, he was (and is) widely regarded as "the greatest logician since Aristotle" (Oppenheimer said that, Aristotle was the father of logic). Einstein said that the only reason he showed up for work at the IAS in Princeton in his last years was so he could walk home with Godel. In his spare time, Godel was the first person in the world to show how Einstein's equations allowed for the possibility of time travel. He did this, not to show how to travel through time, but to show that time has no real existence, it is instead a consequence of the way in which our minds are organized.

: So much for the pedigree, here's some ideas from the book: the existence of an immortal soul can and will be proved scientifically, computers can never be conscious, and mathematical theorems have an existence every bit as real as the chair you are sitting in.

: I was an agnostic before I read this book. Now I know that "mind" and "soul" are just two words for the same thing. Godel is the smartest man that ever lived, and this book contains some of his most interesting ideas in a (reasonably) accessible form. Don't expect to understand more than 10% of it the first time you read it, I have been reading it for years and understand maybe a quarter of it.

Meet Gödel the philosopher.......2000-06-07

Many mathematicians know about Gödel's famous theorem. But very few know about Gödel the man. Through this book, we come to know the man, especially Gödel the philosopher.

Through this book we find out that although Gödel and Einstein were close friends, Gödel, unlike Einstein, shunned public debate. He held philosophical views which he knew would be very controversial if he were to publicize them, and he greatly disliked publshing anything he could not prove rigorously. Accoringly, he instructed his biographer to publish these viewpoints only after his death.

This book contains hundreds of quotations from Gödel's conversations with the author. Fortunately, the author left in quotations that he he said he did not understand, trusting that others might.

Here are a few quotes:

"Consciousness is connected with one unity. A machine is composed of parts."

"The brain is a computing machine connected with a spirit."

"Materialism is false."

"Our total reality and total existence are beautiful and meaningful . . . . We should judge reality by the little which we truly know of it. Since that part which conceptually we know fully turns out to be so beautiful, the real world of which we know so little should also be beautiful. Life may be miserable for seventy years and happy for a million years: the short period of misery may even be necessary for the whole."

If you find Gödel's theorem interesting, I hope you will read this book and found out more about the man behind the theorem.

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An Introduction to Gödel's Theorems (Cambridge Introductions to Philosophy)
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Manufacturer: Cambridge University Press
ProductGroup: Book
Binding: Paperback

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In 1931, the young Kurt Gödel published his First Incompleteness Theorem, which tells us that, for any sufficiently rich theory of arithmetic, there are some arithmetical truths the theory cannot prove. This remarkable result is among the most intriguing (and most misunderstood) in logic. Gödel also outlined an equally significant Second Incompleteness Theorem. How are these Theorems established, and why do they matter? Peter Smith answers these questions by presenting an unusual variety of proofs for the First Theorem, showing how to prove the Second Theorem, and exploring a family of related results (including some not easily available elsewhere). The formal explanations are interwoven with discussions of the wider significance of the two Theorems. This book will be accessible to philosophy students with a limited formal background. It is equally suitable for mathematics students taking a first course in mathematical logic.

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Logic and Structure
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From the reviews: "A good textbook can improve a lecture course enormously, especially when the material of the lecture includes many technical details. Van Dalen's book, the success and popularity of which may be suspected from this steady interest in it, contains a thorough introduction to elementary classical logic in a relaxed way, suitable for mathematics students who just want to get to know logic. The presentation always points out the connections of logic to other parts of mathematics. The reader immediately see the logic is "just another branch of mathematics" and not something more sacred." Acta Scientiarum Mathematicarum, Hungary

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Enthusiastically recommended for anyone interested in following van Heijenoort's challenging life

From Trotsky to Gödel
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Manufacturer: AK Peters, Ltd.
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This story of a highly intelligent observer of the turbulent 20th century who was intimately involved as the secretary and bodyguard to Leon Trotsky is based on extensive interviews with the subject, Jean van Heijenoort, and his family, friends, and colleagues.

The author has captured the personal drama and the professional life of her protagonistranging from the political passion of a young intellectual to the scientific and historic work in the most abstract and yet philosophically important area of logicin a very readable narrative.

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Enthusiastically recommended for anyone interested in following van Heijenoort's challenging life.......2007-09-02

From Trotsky to Godel: The Life of Jean van Heijenoort is the true-life biography of a man in a prime position to observe twentieth-century turmoil. He served as a secretary and bodyguard to Leon Trotsky, and was intellectually and politically passionate; in his maturity, he contributed greatly to the realm of abstract philosophical logic. From Trotsky to Godel draws heavily upon interviews with van Heijenoort himself, as well as his friends, family, and colleagues. Enthusiastically recommended for anyone interested in following van Heijenoort's challenging life, and also for students of Trotsky seeking to round out their perspectives on the man and his writings.

Algebraic and Proof-Theoretic Aspects of Non-Classical Logics: Papers in Honor of Daniele Mundici in the Occasion of His 60th Birthday (Lecture Notes in Computer Science)

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Algebraic and Proof-Theoretic Aspects of Non-Classical Logics: Papers in Honor of Daniele Mundici in the Occasion of His 60th Birthday (Lecture Notes in Computer Science)

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Edited in collaboration with FoLLI, the Association of Logic, Language and Information, this book constitutes the third volume of the FoLLI LNAI subline. The 17 revised papers of this Festschrift volume - published in honour of Daniele Mundici on the occasion of his 60th birthday - include invited extended versions of the most interesting contributions to the International Conference on the Algebraic and Logical Foundations of Many-Valued Reasoning, held in Gargnano, Italy, in March 2006.

Daniele Mundici is widely acknowledged as a leading scientist in many-valued logic and ordered algebraic structures. In the last decades, his work has unvelead profound connections between logic and such diverse fields of research as functional analysis, probability and measure theory, the geometry of toric varieties, piecewise linear geometry, and error-correcting codes. Several prominent logicians, mathematicians, and computer scientists attending the conference have contributed to this wide-ranging collection with papers all variously related to Daniele's work.

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