A taste of Rust

No, that's not an example of Gödelian incompleteness; it's just an example of diagonalisation. Gödel's first theorem says specifically that no countably axiomatisable theory can be both consistent and complete; and, indeed, the proof of the theorem is a diagonal argument like that of your books. There are many fundamental results in mathematics -- such as Cantor's, or Church's -- along similar lines, and though related, they are all different results.