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.
Gödel's first published proof of his incompleteness theorem was entirely syntactical. It described a formula, which I'll call G, such that neither G nor ~G is a theorem of a theory he called P, just so long as P is omega consistent. Do you think that proof is invalid? If so, where in that proof does the first error occur?
> 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.
That a consistent set of first order formulae has a model in proved in texts on logic, including Shoenfield's that you are fond of mentioning. Where in Shoenfield's text does the first error occur?
> Note: cGC <-> "There are infinitely many counter examples of Goldbach > Conjecture". > > Any constructive response would be welcomed.
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