Early in the second decade of the twentieth century, Bertrand Russell and Alfred Whitehead published their monumental work "Principia Mathematica". In it, they claimed to have laid out the mathematical foundations on top of which the demonstration of all true propositions could be constructed.
However, Kurt Gödel's milestone publication of 1931 exposed fundamental limitations of any axiomatic system of the kind presented in "Principia Mathematica". In essence, he proved that if any such axiomatic system is consistent (i.e., does not contain a contradiction) then there will necessarily exist undecidable propositions (i.e., propositions that can not be demonstrated) that are nevertheless true. The original presentation of Gödel's result is so abstract that it is accessible to only a few specialists within the field of number theory. However, the implications of this result are so far reaching that it has become necessary over the years to make Gödel's ideas accessible to the wider scientific community.
In this book, Nagel and Newman provide an excellent presentation of Gödel's proof. By stripping away some of the rigor of the original paper, they are able to walk the reader through all of Gödel's chain of thought in an easily understandable way. The book starts by paving the way with a few preparatory chapters that introduce the concept of consistency of an axiomatic system, establish the difference between mathematical and meta-mathematical statements, and show how to map every symbol, statement and proof in the axiomatic system on to a subset of the natural numbers. By the time you reach the crucial chapter that contains Gödel's proof itself all ideas are so clear that you'll be able to follow every argument swiftly.
The foreword by Douglas Hofstadter puts the text of this book into the context of twenty-first century thinking and points out some important philosophical consequences of Gödel's proof.
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Kindle formatting needs work.
Interesting and well-written book. The other reviews here do a great job discussing the content of this book, so I will not linger on that aspect.You should note that the Kindle Edition had some confusing formatting. For example, here is a screenshot of a page ([...]) where there is a severe lack of formatting for exponents. Should at least read "2^8 x 3^(11^2) x 5^2 x 7^(11^2) x 11^9 x 13^3 x 17^(11^2)", so that it is at least marginally easier to discern where the exponents are. Yes, you ought to be able to figure it out with a moment's thought if you're reading the text that accompanies it, but I thought it was a little disappointing and that it warranted some mention in a review. This has made me cautious of buying any further math or math-related books for my kindle.
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