Subject:
Godel’s Proof of that you cannot axiomatize arithmetic in a manner expressible through that calculus.

Notes:
Godel used a very interesting system to assign his Godel Numbers, to create a unique numeral representing any given equation which is reconstructible. He did this to allow the construction of the rest of the proof.
The book was a great exercise in understanding the nature of proofs and logic. A good experience all in all.