Godel's Incompleteness Theorems (Oxford Logic Guides, No 19)

Rating:
Summary: Mainline Incompleteness with this Book!
Review: I highly recommend this title because it supplys all the necessary proofs for a nuts and bolts understanding of incompleteness, including incompleteness proofs for Peano arithmetic and the unprovability of consistency.

This title is a difficult read but the only prerequisite is a familiarity of first-order logic equivalent to a one semester college course.

A lot of the proofs are based on new material and are easier to understand than the original work by KG.

An added benefit is the exercises. They are not impossible and aid in one's understanding.

This book is well worth the work in demands.

Rating:
Summary: Finally -- Straight Talk About Incompleteness!
Review: Well. This is the book. Read this instead of, or before you read Goedel's paper. Within 20 pages you will know the 'trick' that Goedel used. It's a beauty, but it is far easier to see it under Smullyan's tutelage than by coming to the classic paper cold, since Goedel uses a more difficult scheme to achieve his ends. Much work has been done since 1931, and we get the benefit of the stripping-down to essentials that such as Tarski (and Smullyan himself) have contributed.

The book has much of interest to those who wish to pursue the subject of the incompleteness and/or consistency of mathematics, or to come at Goedel from a number of angles. For me, though, the first 3 chapters were enough. I just wanted to find out how K.G. did what he did. Now I know, and I know where to go if I need even more.

The exercises are helpful to keep you on track and test your understanding. They also contribute materially to the exposition. A stumbling-block for many readers will be the extremely abstract nature of the discussion, and the new notations and definitions that constantly come at one. Viewing numbers as strings and strings as numbers (and knowing when to switch from one view to another) will be confusing at first. This is the hard part: what Goedel did, in essence, is demonstrate that one can view proofs in two ways ' as numbers, and as strings of characters. As in viewing an optical illusion, it is sometimes tough to hold the proper picture in mind.

Smullyan's book 'First-Order Logic' is enough preparation for this work. One must here, even more than there, keep straight the difference between the 'proofs' that are part of the subject matter (and so are strings of characters), and the proofs we go through that verify facts about these strings. Before we started reading this book, of course, we had some informal sense that we were going to prove something about proofs. What we are REALLY doing, though, is proving something about 'proofs'. You get the picture. Goedel must have been a lot of fun at parties.

Rating:
Summary: Finally -- Straight Talk About Incompleteness!
Review: Well. This is the book. Read this instead of, or before you read Goedel�s paper. Within 20 pages you will know the �trick� that Goedel used. It�s a beauty, but it is far easier to see it under Smullyan�s tutelage than by coming to the classic paper cold, since Goedel uses a more difficult scheme to achieve his ends. Much work has been done since 1931, and we get the benefit of the stripping-down to essentials that such as Tarski (and Smullyan himself) have contributed.

The book has much of interest to those who wish to pursue the subject of the incompleteness and/or consistency of mathematics, or to come at Goedel from a number of angles. For me, though, the first 3 chapters were enough. I just wanted to find out how K.G. did what he did. Now I know, and I know where to go if I need even more.

The exercises are helpful to keep you on track and test your understanding. They also contribute materially to the exposition. A stumbling-block for many readers will be the extremely abstract nature of the discussion, and the new notations and definitions that constantly come at one. Viewing numbers as strings and strings as numbers (and knowing when to switch from one view to another) will be confusing at first. This is the hard part: what Goedel did, in essence, is demonstrate that one can view proofs in two ways � as numbers, and as strings of characters. As in viewing an optical illusion, it is sometimes tough to hold the proper picture in mind.

Smullyan�s book �First-Order Logic� is enough preparation for this work. One must here, even more than there, keep straight the difference between the �proofs� that are part of the subject matter (and so are strings of characters), and the proofs we go through that verify facts about these strings. Before we started reading this book, of course, we had some informal sense that we were going to prove something about proofs. What we are REALLY doing, though, is proving something about �proofs�. You get the picture. Goedel must have been a lot of fun at parties.