On 18/10/2013 5:34 AM, Peter Percival wrote: > 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.
What definition of "invalidity" were you referring to _here_ ? Mine? > > Rosser's strengthening of Gödel's incompleteness theorem omits "omega".
So in his theorem, Rosser did _not_ include the assumption something _similar to my_ "adequate enough to describe the concept of the natural numbers" for his underlying T?
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