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This book, "Godel, the life of logic", is a book that has a lot of junk bio, but it also has ideas that one is compel to reread. For example the Godel Number and Turing Machine. It does draw the reader with interest. I thought I knew everything about logic (Boolean logic mostly) but I was not properly informed.

The whole book drives with the idea of truth (semantics) and proof (syntax). Syntax is the logical machinery that produce all the mathematical (logical) statements. It also can be said that syntax is the formalization. There is no mathematical intuition that allows one to see all the mathematical statements. Therefore all the syntactical statements are statements proved (or disproved). So we can map out what are proved and disproved. There will be statements that will be not ( proved and not proven).

There were sections about AI and there are bio that are equally uninteresting (except Godel apparently starved himself which seems so appropriate of a logician.) The second to the last chapter "The Complexity and the Complexity". A chapter on computerized number theory is very interesting. I never found number theory interesting in school but this one was a kicker. For example, Godel's Theorem (Diophantine Equation Version)---there exist a Diophantine equation that has no equation but no theory can prove this. Diophantine equations are equations that have two or more variables, and it should have integer solutions. For example, x^2+y^2=z^2 can have solution of x=3, y=4, z=5 (for z being the length of the slope of a Pythagorean triangle).

As I said, I am going back and reread sections of this book: Godel Number and Turings Machine, etc...If you are new to modern logic this will be an eye opener.

After rereading Godel, Escher, and Bach three times, I may have reached 15% comprehension but the topic still interested me. This book on Godel is designed well for non mathematicians or PhD logicians. It is well written. The authors explain the concepts clearly, provide intersting context and background and avoid condescension. Godel's work reminds us of the limits of our certainty and would be useful for both atheists and fundamentalists to read.

This book seems to be put forth as a biography of Godel--at least, that's what I thought I would be getting. As a physics and mathematics teacher, I am very interested in this man who revolutionized 20th century thought. However, only a portion of this already very slight book, is biographical material about Godel. So, if you are looking for a real biography, look elsewhere.
Instead, this book briefly covers Godel's life, briefly covers his work, and briefly covers a few of the effects Godel has had on current thought. In that sense, it is not bad. A person who understands very little about modern mathematical thought but has an interest might find this book digestable and learn a bit. Others are not going to enjoy this book. People with strong backgrounds in mathematics are going to find it too weak and people with weak backgrounds in mathematics are going to find most of it indecipherable.

This book gives clear insights in the life of one of the greatest mathematicians of the 20th century. Where Gödel’s intellectuality is easily noticeable, his anxious nature is hidden under a layer of introversion. Both the biographical details as well as the comprehensible explanation of Gödel’s groundbreaking accomplishments in mathematical logic and philosophy make this book a delight to read.

This book is a wonderful story of mathematical heroism. One in which a shy young mathematical genius, Kurt Godel, takes on, and then at least metaphorically, slays the mathematical giant of his day, David Hilbert. Hilbert for most of the 20th Century was the acknowledged, crowned and unchallenged super-genius and father of all mathematics. And as part of the honor for being the reigning mathematical genius, at the end of each decade of his professional life, for nearly a half century, Hilbert was allowed to promulgated from "on high" a series of fundamental unsolved questions about mathematics and logic, questions that if solved and found to be untrue, could uproot and overturn the very foundation of our system of mathematical logic. One such question Hilbert posed was whether it was possible to prove every true mathematical statement? What Hilbert was looking for with this question was a kind of truth machine, in which one could just feed in a statement; turn the crank, and viola, out would come a true or false answer, but never both.

Up to the intellectual challenge to the mathematical gauntlet thrown down by Hilbert in 1928, Kurt Godel, the mild-mannered unassuming member of the famous Vienna Circle, solved one of Hilbert's outstanding foundational problems, and thus proved in his famous incompleteness theorem that our system of logic does indeed have holes in it. That is to say, he showed that the system of logic upon which arithmetic depends is inconsistent or indeed that it is logically flawed; so that when the crank to Hilbert's machine is turned, one could never be sure that only "true" or "false" answers would result. As in Russell's famous paradoxes, sometimes the answers could be both "true" and "false."

Godel was raised in the heady intellectual milieu of Franz Kafka, Karl Popper, John Von Neumann, Bertrand Russell, Albert Einstein and Ludwig Wittgenstein. And as a member of the prestigious Vienna Circle, got to sit at the table with, and was a disciple of, the erratic but brilliant philosopher and logician, Ludwig Wittgenstein. Although Wittgenstein believed that there was a logical mirroring between the facts of language and the facts of reality, and that the representation of the world in thought is made possible by logic through language, he still did not believe that the parallel between "facts" and their representations as "language" captured all there was to reality. And as it turns out, it was precisely this gap in Wittgenstein belief that would prove to be the sliver of logic that Godel needed to solve one of the most famous problems posed by David Hilbert. It was this idea of Wittgenstein's about the parallelism between logic and language that Godel gave elaborate and deep mathematical form, and used it to prove, as Hilbert's questions had suggested, that our system of logic did indeed have a hole at its very center.

Both a proper statement of the problem and a proper framework for setting up the solution to Hilbert's problem are difficult intellectual enterprises on the road to understanding how Godel's theorem solves the problem. Several books have been devoted to trying to make both aspects clearer and more accessible to the general reader. Each has had its good and bad points. The one by Rebecca Goldberg, for instance, does an excellent job of restating and better defining the problem, but in my view fails to make the details of the solution clear enough to stick to an ordinary brain. Douglas Hofstadter, on the other hand, in his tour de force, "Godel, Escher and Bach," errs in the opposite direction. He uses Disney like characters and inventive games as props that do all of the heavy lifting, and thus an excellent job of explicating the details of Gödel's proof. But again, in my view, he spends too little time in properly setting up and framing the problem so that the details are easily grasped by the general reader.

To the present authors' credit, they seem to have benefitted from and to have drawn inspiration from both of these earlier efforts, and as a result, have gotten both aspects just about right. They thus have succeeded admirably here at both, defining the problem and explaining the details of the proof of Gödel's theorem. But they do not stop there, just with a proper analysis of the theorem, they go on to explain why Gödel's theorem has proven to be important to further developments in mathematics, computer science, artificial intelligence and logic as well as analytic philosophy.

Put in its simplest and perhaps most compact form, Gödel's Theorem can be stated thusly: For every consistent formalization of arithmetic, there exist arithmetic truths that are not provable within that formal system. On page 51 of this book, the authors give a skeletal outline of how Godel went about proving his theorem. In my view, that outline is difficult to improve upon, so I will end this review by stating the author's steps in my own words:

1.Godel first developed a coding scheme (using numbers to represent statements about numbers) that translated every logical formula in arithmetic into a mirror image statement about the natural numbers. That is, numbers coded so that they can be used as a language to talk about numbers.

2.He then replaced the notion of "truth" with the idea of "provability," preparing the ground to eliminate number language statements that could lead to contradictions such as those by Russell and others that have long bedeviled mathematics as well as our system of logic.

3.After this he showed that logical sentences have arithmetical counterparts (Wittgenstein's parallelism) called Godel sentences, G. He then proved that the Godel sentence G must be true if and only if the formal system is consistent.

4. Then he showed that even if additional axioms were added to the existing system to form a new system, there would still be un-provable Godel sentences within the new system.

5.Finally, using his coding system, he showed that the statement "arithmetic is consistent" is not provable, thus showing that arithmetic as a formal system was too weak to prove its own consistency. QED. Five stars.