an exceptionally clear exposition of Godel's Incompleteness Theorem, Godel's Proof by Ernest Nagel

I gave this a try. Never finished it though.

Penrose: The Emperor's new mind... Douglas Hofstader: Godel, Escher, Bach: An eternal golden braid

I have the Hofstader tucked away somewhere in my efficient filing system. It is the source of my present incomplete understanding. I shall endeavour to find it and I will scope out the Penrose.

My present understanding is basically that any system of thought can produce statements like "this statement is a lie" and that, while these statements can be resolved by building some metasystem that includes the offending system as a subset, that metasystem will also be capable of generating similar statements. I take this to mean that we are not infinitely intelligent.

PS
According to Ænigma the book is Douglas Hog: Godfather, Escherichia, Bach: An eternal golden braid

PPS
I love Ænigma's correction of her own name.