Isn’t the order of a proposition included in its Gödel number?
Each proposition is assigned to an increasing prime power, and the increasing list of primes has total order, such that swapping propositions yields a distinct Gödel number.
I think what ProfHewitt means here (based on other writing I've found, as he is frustratingly low on details in these conversations) is "order" in the sense of "first order logic", "second order logic"; not in the sense of "first proposition, second proposition" etc.
His claim is that the proposition "This proposition is not provable", formalized as "P equivalent to P is not provable" is not well formed in a typed logic, as "P"'s type is first-order, while "P is not provable"'s type is second-order. Therefore, his claim is that the proposition is simply not well-typed and therefore not interesting. Godel's proofs were discussing an untyped logic, but according to Hewitt that is not an accurate representation of mathematics.
I don't think anyone in the space agrees with him, though, as far as I could tell from some cursory reading.
Could you provide two statements with equal Gödel numbers and the same characters in different orders, and a detail of how you compute the Gödel numbers?
If two statements have the same characters in the same order, how are they not the same statement? And why would that make Gödel’s statement invalid in Principia Mathematica?
Each proposition is assigned to an increasing prime power, and the increasing list of primes has total order, such that swapping propositions yields a distinct Gödel number.