[Cooker]

But the one key point of Godel's theorem is the type of system. I believe it must have an infinite recursive structure to fall into the class of theories that must be incomplete. If the final theory doesn't have such a structure, then it is possible that it could be complete (like euclidean geometry).

I personally feel as though physics is being overly dominated by a search for a final theory, but I think a final theory is still a possibility (although I entertain many ideas about possibilities here as well). If it is to be complete, then it will have to be much simpiler than the arithmetic of the natural numbers I agree. However, I do not agree that a final theory must be more complex than the arithmetic of the natural numbers, so I think completeness is still a perfectly possible choice. I think this would suggest that the natural numbers are an invention of man and not of nature.