On 10/20/2013 1:55 AM, Nam Nguyen wrote: > On 04/10/2013 9:47 PM, Nam Nguyen wrote: >> Two major theorems Godel's Incompleteness are: >> >> - Incompleteness: Any formal system T that is consistent _and_ adequate >> enough to describe the concept of the natural numbers, would have G(T) >> as a statement that is true but not provable in T. >> >> - Completeness: Any consistent formal system has to have a model. >> >> On the Incompleteness, since the requirement that T be _informally_ >> adequate enough to describe the concept of the natural numbers is _not_ >> a syntactical notion [as that of a T's consistency], it's logically >> invalid to assume that T always be syntactically consistent, simply >> because we _informally assume_ T adequately describe the concept of >> the natural numbers. QED. > > More to the point, on Incompleteness, the 2 assumptions > > C1 = T is consistent. > C2 = T is adequate enough to describe the concept of the natural > numbers. > > are distinct as well as necessary. But since it's impossible to > logically guarantee that if T satisfies the _informal_ notion of C2, > T would also satisfy the _formal_ notion of C1, Incompleteness is an > invalid theorem. > > Note: by "formal" notion here, we'd mean a notion where the concept > of rules of inference is required, but also where the concept of the > natural numbers (or related notions such as recursion, induction, > language structure, etc...) is _NOT_ required. >
Well, one interpretation of the problem is that anything based upon "what is prior" and "what is posterior" with a first step is interpretable relative to the ordinal sequence of natural numbers.
Rules of inference are stepwise syntactic transformations.
How do you formulate a deductive theory which is not interpretable in relation to the ordinal sequence of natural numbers from which you wish to assert independence?
Aristotle has a discussion concerning four forms of priority (which, unaccountably, turns into five forms by the end of the passage). Deductions and ordinal sequences are bound in this way and cannot be undone.
Your second paragraph appears to ask for a distinction which is indemonstrable and inconceivable.
Incompleteness merely reflects that numbers and deductions cannot be placed into the relation of priority with respect to one another.