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. > > 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.
For PA, Gödel's incompleteness theorem says that, if PA is omega consistent, then there is a sentence G, say, in the language of PA such that neither PA |- G nor PA |- ~G. If that is invalid then there is no such G, so, in particular one of cGC and ~cGC is a theorem of PA.
Rosser's strengthening of Gödel's incompleteness theorem omits "omega".
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