@user170039: I've read about 20% of the pdf so far. I wish to start mentioning key points so that I don't forget them later. The first point is that Wittgenstein is being very imprecise in what he is said to have written (page 10). I will fault him for at least that, even if one argues that he actually had complete grasp of the incompleteness theorems. Specifically, he implies that Godel's "true" means "proved in Russell's system".

However, he's wrong to say "Gödel was barred by virtue of the logical grammar of mathematical proposition from claiming that he had constructed identical versions of the same mathematical proposition in two different systems." Godel didn't do such a thing, as would be clear to anyone who has basic grasp of logic and model theory.

But it is invalid to go from that claim to the general claim that all mathematical statements are of the same nature.

@user170039 Thanks, but I think I'll pass. Unless someone shows a flaw with the incompleteness theorems (which I've myself proven), I just don't see a need to know what exactly the authors of PM were thinking. If their system is really not susceptible to the incompleteness theorems, then it necessarily must either be imprecise or unable to prove at least one true Π1-sentence of arithmetic, which is quite serious a problem, to say the least.

Although unrelated, recently while reading a significant amount of literature on Russell's logic (or more specifically the logic of Principia Mathematica), I found that your viewpoint as stated in the following,