The Incompleteness Phenomenon [Hardcover]

Martin Goldstern (Author), Haim Judah (Author)

Book Description

ISBN-10: 1568810296 | ISBN-13: 978-1568810294 | Publication Date: June 15, 1995

This introduction to mathematical logic takes Gödel's incompleteness theorem as a starting point. It goes beyond a standard text book and should interest everyone from mathematicians to philosophers and general readers who wish to understand the foundations and limitations of modern mathematics.

Product Details

• Hardcover: 218 pages
• Publisher: A K Peters/CRC Press (June 15, 1995)
• Language: English
• ISBN-10: 1568810296
• ISBN-13: 978-1568810294
• Product Dimensions: 9.4 x 6.3 x 0.8 inches
• Shipping Weight: 1.1 pounds (View shipping rates and policies)
• Average Customer Review: 2.8 out of 5 stars  See all reviews (4 customer reviews)
• Amazon Best Sellers Rank: #2,290,947 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

14 of 14 people found the following review helpful:

4.0 out of 5 stars A good simplification for teaching, October 2, 2000

By 

Colin McLarty (Chardon, OH USA) - See all my reviews

This review is from: The Incompleteness Phenomenon (Paperback)

This is an interesting simplification of Goedel's first incompleteness theorem. The book assumes there is a standard model of the Peano axioms, so that in effect it assumes the axioms consistent and even true. Then it shows nonetheless the axioms cannot decide every sentence. This brings out the main point of incompleteness, I think. But it is far weaker than Goedel's proof in two ways: it uses stronger assumptions, and it proves incompleteness only for the standard Peano axioms. There is a brief discussion of how this kind of proof would work for any consistent extension of the Peano axioms but I did not find it very helpful to my class. Of course the assumptions pay off in a very much quicker proof and much less concern with syntax.

In fact, the main problem with the book is that the assumptions are never made quite clear. The authors say several times that their result is weaker than Goedel's, but never say why. A discussion of this somewhere in the book would be helpful--both to students and profs. Presumably they do not use the full strength of a standard model of the Peano axioms, but it would be a chore to go through and see just what they do use.

I taught this book as a one semester course for students who had previously seen predicate logic in an intro course. To fit it into a semester I skipped the chapter on model theory (not needed for the incompleteness theorem) and the material at the end on recursive functions. The book gives a very pretty account of induction, stressing from the start that the natural numbers are just one case of an inductive structure. This made later inductions on, say, well-formed formulas, very clear to the class. The students got the compactness theorem very easily, as they had not in other class I've taught from other books. The short account of non-standard models for arithmetic is helpful in showing that Goedel's theorem is *not* about whether the Peano axioms say all there is to know about arithmetic--the fairly simple compactness theorem already shows no first order theory can do that. I expect to use this book again the next time I teach the subject.

4 of 7 people found the following review helpful:

2.0 out of 5 stars Nope, didn't like it., September 20, 2003

By 

Jason T (Canada) - See all my reviews

This review is from: The Incompleteness Phenomenon (Hardcover)

This book is an introduction to mathematical logic, covering the syntax and semantics of propositional and first-order logic, the Hilbert-style proof system and its completeness, some model-theoretic material, and Godel's (first) incompleteness theorem. Its more formal and rigorous than most introductory books, which is the style I prefer, but I was left feeling unsatisfied with the book. It was hard to nail down exactly what I didn't like about it, but what I came up with is this: although the theorems and proofs are ok, considered one at a time, the overall perspective of what's going on and how things relate to each other was left hazy. Perhaps better exposition and historical background would correct this, but I found the book unsuitable for self-study for a beginner. This was where I first learned Godel's incompleteness theorem, and even though the version presented is particularly weak (Peano arithmetic is incomplete), I was left confused about the significance of the theorem and exactly what assumptions were used in the course of the proof. I see now that their attempt at simplification is what led to my misunderstandings. If you're looking for a good general mathematical logic book, I seriously recommend you get Enderton instead (see my reviews). If you want a book focused on the incompleteness theorems get Smullyan's excellent GIT.

1 of 3 people found the following review helpful:

3.0 out of 5 stars Corrected second printing exists, September 20, 2007

By 

Dana Roth "dzrlib" (usa) - See all my reviews
(REAL NAME)   

This review is from: The Incompleteness Phenomenon (Paperback)

In the words of a faculty member "this first printing is so riddled with typos and grave mathematical errors that you would have to include a multi-page errata sheet with it to make it usable." There is no indication in any listing for the book that a corrected printing exists. We called the publisher to insure we were getting the second printing, but at least one patron lamented that they had purchased a copy online and ended up with the first printing. The only indication of changes in the 2nd printing is a single sentence on page xiii thanking the faculty member for submitting corrections over the first printing. [e-mail] 9/20/07