Incompleteness: The Proof and Paradox of Kurt Gödel

This book is about the life and work of Kurt Gödel. And what better way to start a book on Gödel’s Incompleteness theorems, by adding, as a first note in your book, a self-referencing note? That is what Rebecca Goldstein does in “Incompleteness”, by ending her first note with an additional note that there are two types of notes used in her book, namely footnotes and endnotes.
But in her charming book she does much more than only adding notes to some theorems, by explaining that self-reference, and the logical paradoxes it leads to, are really at the heart of Gödel’s proof of his incompleteness theorems. She outlines the proof very well leaving out the technical details, but in such a way that an interested layman can still appreciate the crux of it.
Next to her explanation on the incompleteness theorems, she does a great job by putting the work done by Kurt Gödel in its historical context and describing the man behind the theorems. And what a man Gödel was: closely befriended with Albert Einstein, admired by Alan Turing and Roger Penrose. And “opposed” by men like Ludwig Wittgenstein and Bertrand Russel.
When you strip down reasoning from all its softer arguments and intuitions, in order to build a solid basis for your mathematics, Gödel proved that your basis will be either inconsistent or incomplete. That is, either being able to proof any theorem to be true (or false, as you like) or not being able to proof some true theorems where the truth follows from an "outside-the-system" reasoning.
According to Roger Penrose Gödel’s Incompleteness theorems are implying that Artificial Intelligence, that is machines, will never be able to think as humans do. Gödel added the more precise statement that either we humans are no machines or that we are deluded machines thinking we are not. Which is a perfect Gödelian phrasing since machines cannot think.

Jeroen Bos-mulder's Reviews: Incompleteness > The Proof and Paradox of Kurt Gödel