The Complete Collection of Incompleteness
Gödel's Incompleteness Theorem has been one of the most requested videos on Numberphile.
The task of explaining it was taken up by none other than Professor Marcus Du Sautoy (a household name in the UK, at least when it comes to mathematics!)
We posted three videos - a main explanation plus two lots of "extra footage" .
Professor du Sautoy covers the topic in his latest book too.
In the US it is called The Great Unknown - http://amzn.to/2sfkWpb
In the UK it is called What We Cannot Know - http://amzn.to/2r5yztp
More of the professor's books: http://amzn.to/2ryEk4r