“semi” rigurously is what spooked me. The thing with Gödel’s proof is that it is very hard to do rigurously...
[1] https://www.goodreads.com/book/show/1556746.Computability_an...
It's hard to explain, because the idea of encoding statements about a system in terms of the system itself is weird if you haven't seen it before.
That said it can be fully covered in a single semester of undergraduate study, so its not that hard. There are many theorems that couldn't be fully proved in a semester.
“My nose will get longer”