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.
> > On the Completeness, since it's still entirely logically possible that > it's impossible to know the truth value the formula cGC, or ~cGC, > (defined below) in the natural numbers, it's still entirely logically > possible that it's impossible to have a model for PA + cGC, or PA + > ~cGC. Hence it's logically invalid to assert in meta level that a > consistent formal system must necessarily have a model. > > Note: cGC <-> "There are infinitely many counter examples of Goldbach > Conjecture". > > Any constructive response would be welcomed. >
-- ----------------------------------------------------- There is no remainder in the mathematics of infinity.