Ok, why couldn’t set theorists defeat Gödel by simply requiring that a symbol only means one thing?
For example, we define the operator “+” to mean {what it means} and no other symbol can mean “+”, nor can “+” mean anything other than how it is defined. Wouldn’t this eliminate the ability to construct a Gödel number?
I think the Gödel numbers were just a method he used to find the unprovable theorems more easily. The flaw in the system, if I remember my Hofstadter, is that formal logic is complex enough to allow statements to make statments regarding their own truth. I don’t think Gödel numbers are essential to that.
But I’m getting confused and I would like an answer myself. 
The real trick of arithmetization is that a number can represent a statement or a proof. As long as you have ordinary arithmetic on numbers there is no way you can formally ban what amounts to a concept. I.e., you would have to ban “+” and company entirely. You can easily restrict a logical system to prevent the use of numbers to make statements about statements. But all such known systems are too weak to make what are regarded as important and useful mathematical statements (and could still be incomplete anyway). In Computer Science we deal with a lot of different restricted logical systems that are useful for certain situations but are not very general.
Goedel’s biggest result is that a system is either too weak to be considered “Mathematics” or sufficiently strong to be incomplete. (By “Mathematics” I mean whatever you might define as your favourite accepted interpretation of set theory or logic.)
Hmm, let me summarize this, all systems can divided into 4 categories:
Note that there is no “Complete but generally useful” category. No matter how you define “+”, it can’t happen. It’s an “impossible task” like trisecting an angle with straight edge and compass.
It has to do with the notion of theoremhood in predicate logic. When I say sym(Kx ® Nx)[/sym], there is absolutely no meaning inherent to the letters N and K. Actually, there’s no meaning inherent in any of the symbols, but we’ve agreed on them. In order for this statement to be true, it has to hold no matter what meaning we assign to N and K. So if the set theorists (logicians, actually, but I’m not nitpicking) did this, they’d be crippling formalized mathematics.
That should read “In order for this statement to be a theorem…”.