Has anybody read each and every line of the english translation of the 1931 Godel paper (from page 40 to the end) ?

https://homepages.uc.edu/~martinj/History_of_Logic/Godel/Godel%20%E2%80%93%20On%20Formally%20Undecidable%20Propositions%20of%20Principia%20Mathematica%201931.pdf

I tried once, but the notation is so far from the modern notation, and the setup is so strange (use of abitrary order formulas) that I found it quite difficult to follow.

If somebody managed to get to the end, I have the following question for him (her) : is this proof purely syntaxic ? does it avoid the completeness theorem ?

The proofs I read are all use the completeness theorem at some point, so I wonder what a purely syntaxic proof would be like, if it happens to exist.