Ernest Nagel and James Newman describe for the layman the famous “Incompleteness Theorem” of the mathematics genius Kurt Gödel. It was updated and edited in 2001 by Douglas Hofstadter, who wrote the foreword.

As the front flap says, “First published in 1958 and in print continuously in ten languages, this highly popular, seminal work offers every educated person with an interest in mathematics, logic, and philosophy the opportunity to understand a previously difficult and inaccessible subject.” The subject is Gödel’s “Incompleteness Theorem.” Gödel’s 1931 paper “On Formally Undecidable Propositions of Principia Mathematica and Related Systems” was written for mathematicians/logicians, but the original of ”Gödel’s Proof” was written for anyone. As ”Gödel’s Proof” is the book that determined Hofstadter’s future in math, I bought it because I had so greatly enjoyed Hofstadter’s Pulitzer-Prize winning ”Gödel, Escher and Bach: an Eternal Golden Braid’ – book book>149.

However, since I had already read book 149 as well as Roger Penrose’s books 121 Shadows of the Mind: A Search for the Missing Science of Consciousness and 150 The Road to Reality: A Complete Guide to the Laws of the Universe, I found little new in this book. But I can easily understand why Hofstadter wanted to improve on it.

I found a very unexpected new thing in Hofstadter’s updated version (this book 160), which I quote: “Meta-mathematical arguments establishing the consistency of formal systems such as PM have, in fact, been devised, notably by Gerhard Gentzen, a member of the Hilbert school, in 1936, and by others since then. These proofs are of great logical significance, among other reasons because they propose new forms of meta-mathematical constructions and because they thereby help make clear how the class of rules of inference needs to be enlarged if the consistency of PM and related systems is to be established.” [In this quote PM means Principia Mathematica and meta-mathematics refers to statements about mathematics – in contrast to statements within mathematics.] This means that the attempts of logicians to “get around” Gödel’s “Incompleteness Theorem” continue, so a formal system like Bertrand Russell’s and Alfred North Whitehead’s PM may yet “be established,” which would indeed be “of great logical significance.” I wonder why neither Hofstadter nor Penrose mentioned this in their books above. At any rate, if one wants to really digest Gödel’s “Incompleteness Theorem” in a book focused solely on that topic, book 160 is a sure way to do so, without all the “extra stuff” in books 149, 121, or 150.