If I take "1 is even", "1 is odd" and "no number can be even and odd" as my axioms, then there is obviously a problem, but of my own doing.
Before Godel's time, people just didn't have enough experience with computers to realize that you have to deal with code injection attacks every time you try to build a powerful platform of any kind. In this case, Godel Numbering is the hack that allows code injection into a formal system that's supposed to just be highly insightful about properties of the infinite world of natural numbers.
engineered than its predecessor by Frege because it has orders
on propositions. Because of orders on proportions, PM does
not allow the [Gödel 1931] proposition I'mUnprovable.
Furthermore, adding the proposition I'mUnprovable would
make PM inconsistent.
The Gödel number of a proposition in PM is itself
"incomplete" because it *doesn't* include the order of the
proposition. Allowing its Gödel number to represent a
proposition is indeed a kind of "code injection" attack,
which if allowed would make PM inconsistent.