Jim Choate says: > Godel's does -not- say mathematics is incomplete, it says we can't prove > completeness -within- mathematics proper. To do so requires a > meta-mathematics of some sort.
You are mixing up what Godel says about proving consistency within a system, and his incompleteness theorem. Godel most certainly DOES prove that mathematics is incomplete. >To write a string down to feed to your truth engine is one thing, to be > able to write it in either the 'true' or 'false' list is something > entirely different. Nobody cares about the first part, they care a great > deal about the second. >And no it won't be 'true for some, false for some'. The actual content of > the symbols is of -no interst-. We are trying to determine if the string > is legitimate within the axioms and their grammer, not it's absolute > context sensitive result. Here you are wrong. The content of the symbols is very important. The study of mathematical logic, including Godel, depends on comparing semantic truth (validity in models) to syntactic truth (provability). You are only dealing with the syntax. "Completeness" means that every _valid_ formula can be _proved_. The only connection I see between completeness and what you said about "writing down" every "true" string is this: If the set of valid formulas in a system is not recursively enumerable, then the system cannot be complete. This is true of arithmetic. But this is not the definition of "complete." There are systems that are incomplete for other reasons, even over finite models. And there are infinite systems (e.g. the first-order logic) that are complete, but they are not sufficient to describe all of arithmetic.
