# Customer Review

on December 24, 2001
This is a remarkable book. It examines in considerable detail Godel's proof, a mathematical demonstration noted for its difficulty in its novel logical arguments. The chapter topics - the systematic codification of formal logic, an example of a successful absolute proof of consistency, the arithmetization of meta-mathematics - appear almost unapproachable. And yet, Ernest Nagel and James R. Newman have created a delightful exposition of Godel's proof. I actually read this book in one sitting that took me late into the night. I simply didn't want to stop; it is really a good little book.
Godel's proof is not easy to follow, nor easy to grasp the full implications of its conclusions. Many mathematical texts, overviews, and historical summaries avoid directly discussing Godel's proof as these quotes indicate: "Godel's proof is even more abstruse than the beliefs it calls into question." "The details of Godel's proofs in his epoch-making paper are too difficult to follow without considerable mathematical training. "These theorems of Godel are too difficult to consider in their technical details here." Such is the common reference to Kurt Godel's milestone work in logic and mathematics.
In their short book (118 pages) Nagel and Newman present the basic structure of Godel's proof and the core of his conclusions in a way that is intelligible to the persistent layman. This is not an easy book, but it is not overly difficult either. It does require concentration and a willingness to reread some sections, especially the second half.
"Godel's Proof" begins with an explanation of the consistency problem: how can we be assured that an axiomatic system is both complete and consistent? The next chapter reviews relevant mathematical topics, modern formal logic, and places Godel's work in a meaningful historical context. Following chapters explain Hilbert's approach to the consistency problem - the formalization of a deductive system, the meaning of model-based consistency versus absolute consistency, and gives an example of a successful absolute proof of consistency.
The plot now begins to twist and turn. We learn about the Richardian Paradox, an unusual mapping that proves to be logically flawed, but nonetheless provided Godel with a key to mapping meta-mathematics to an axiomatic deductive system. (I forgot to explain meta-mathematics; you will need to read the story.) And then we learn about Godel numbering, a mind boggling way to transform mathematical statements into arithmetic quantities. This novel approach leads to conclusions that shake the foundations of axiomatic logic!
The authors carefully explore and explain Godel's conclusions. For the first time I began to comprehend Godel's fundamental contribution to mathematics and logic. I am almost ready to turn to Godel's original work (in translation), his 1931 paper titled "On Formally Undecidable Propositions of Principia Mathematica and Related Systems". But first, I want to read this little book, this little gem, a few more times.
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