For a detailed perspective on how the proof works, I highly recommend Ernest Nagel and James Newman's book Gödel's Proof [0], mentioned in the article. Alternatively, Gödel Escher Bach by Douglas Hofstader is a classic which serves as a great (and more accessible) introduction to the proof [1].
[0] https://www.amazon.com/G%C3%B6dels-Proof-Ernest-Nagel/dp/081...
[1] https://www.amazon.com/G%C3%B6del-Escher-Bach-Eternal-Golden...