Proof sketch of Gödel's incompleteness theorems

In this post, I will follow the outline of the proof given in Section VII of Nagel and Newman's book. Recall that:

• a formal system F is consistent if there is no well-formed formula (wff) w such that F proves both w and ¬ w, and
• a formal system is (semantically) complete if it can prove all of the true wff's that it can express.

Now, Gödel's first incompleteness theorem can be paraphrased as:

Any consistent formal system that is expressive enough to model arithmetic is both incomplete and "incompletable."

and Gödel's second incompleteness theorem can be paraphrased as:

Any consistent formal system that is expressive enough to model arithmetic cannot prove its own consistency.

According to Nagel and Newman, a proof of the first theorem can be sketched in 4 steps: