(deindent) Not even wrong only if you don't understand what he's doing. These are all standard problems for the many millions of people who have tried to make sense of infinite arithmetic at some point in their lives.

The number  ...9997 in Archimedes' system is also known as the number -3, which almost is the "largest prime" in Ludwig's order on the Plutonian integers. You would think ...99998 might be the right one, but that's -2, and it is clearly divisible by two. So maybe ...9997 should be thought of as divisible by three, but dividing by three is different than dividing by two in base-10, because two divides 10 and three does not. So maybe that's a self-consistent definition in a 10-based infinite arithmetic with the order he chooses. Maybe not. I don't know, but neither do you, and to pretend otherwise is insincere. It's not incoherent, just original, and incomplete. The methodology is obviously many, many, painstaking infinite digit calculations, with the extra burden that the rules keep shifting whenever he finds an inconsistency. 

The reason he claims transcendental numbers are incomplete is because, if you accept that the integers are Plutonian, the real numbers are only "finitely" wide. Normally, a real number can be thought of as a map from Z->digits, giving it's digit sequence. Unfortunately, Plutonium's "Z" is just as large as the real numbers, so that a real number is really only a sequence of digits, because a sequence in this view cannot be viewed as a map from Z-> digits. In this view Z is larger than any sequence (the integers are "uncountable"). So the real numbers have "gaps" which would be filled by reciprocals of infinite integers. But large integers don't have reciprocals in Plutonium's system, because he can't make any sense out of bi-infinite sequences:

Numbers like ... 999123.3333..... are not allowed because you can't do multiplication digit by digit.

So he doesn't think of the real numbers as containing the integers, he has two classes of numbers the "Adic integers" which are infinite to the left and finite to the right, and the "real numbers" which are infinite to the right and finite to the left, and it is illegal to add an adic integer and a real number.

The reason Archimedes Plutonium keeps working on the Riemann Hypothesis is because he feels he can say something new about the distribution of primes because he has knowledge of some infinite ones. This is because he can ask questions like "if I choose an infinite integer at random, digit by digit, do I get a prime number?" Those sorts of questions have analytic counterparts about real numbers (complete numbers really). He probably feels the twin-prime conjecture is false, becuase, what's the chance that the infinite integer ....321113 and ....321115 are both prime? He also probably feels that an infinite even number can never be the sum of two primes. These last two statements are both based on my own thinking. He might thinks that there are very many primes at infinity, I don't know for sure.

Look, I am not saying his mathematical musing are not cranckish. But recognize that Archimedes Plutonium is to cranck science as Isaac Newton is to mainstream science. He has seen futher than others even though he stands on the shoulders of no-one.