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

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 y as numbers, and as strings of characters. As in viewing an optical illusion, it is sometimes tough to hold the proper picture in mind.

Smullyanys book yFirst-Order Logicy is enough preparation for this work. One must here, even more than there, keep straight the difference between the yproofsy 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 yproofsy. You get the picture. Goedel must have been a lot of fun at parties.


Mainline Incompleteness with this Book!   March 23, 2000
27 out of 28 found this review helpful

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.


A very good book but requires correction of typos   July 6, 2006
Ng, Yui Kin (Hong Kong)
13 out of 14 found this review helpful

Raymond Smullyan is a logician that I admire much. This book is very good but contains many typos and mistakes. For example, in p.31, the definition of xPy does not work as it is not able to account for 0P305. The definition should be corrected as:
xPy iff There is z not greater than y (zBy and xEz). Similar mistakes can be found elsewhere. And this book thus requires another edition for the coorection of typos and mistakes.


Well Worth The Investment   September 26, 2007
A. Siering (Austin, TX)
3 out of 4 found this review helpful


J Alfonso, in a review of Dover's On Formally Undecidable Propositions of Principia Mathematica and Related Systems, wrote:

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.


It was in that review that he suggested interested readers get a copy of Smullyan's book. So I did. I was hesitant because it was expensive, but in the end if you really do want a complete and thorough understanding of Goedel's proof then this is maybe the most efficient way to go about it.

It is a challenging but accessible book that requires little prerequisite knowledge. A basic familiarity in first order logic is all that is required. And quite frankly if you can't make it through this book then I don't understand how you expect to make it through Goedel's actual proof.

I believe that both Smullyan's book and Goedel's proof can be understood by any diligent student, so don't take my last comment to suggest this it's an impossibly difficult task. Rather what I am trying to suggest is that if you really want to understand Goedel's proof then this book may or may not be easy for you, but it certainly will be tremendously valuable and is worth the effort.