I wrote this article in an attempt to distill the most essential part of the argument of Gödel's famous proof. It is based on my reading and interpretation of Gödel's Proof (by Nagel and Newman) - which I very highly recommmend. The mechanics of Gödel Numbering are super neat in my opinion (and also not that hard to understand), but for the purposes of understanding the main argument, it's enough to consider Gödel Numbering as a 'black box' system for labelling arithmetical statements - kind of like a barcode.
So far I only talk about the First Incompleteness Theorem (concerning the provability of statements). When I get the chance, I will put a follow up article here, about the Second Incompleteness Theorem (concerning consistency of formal systems).