> I believe you can make a consistent fully provable axiomatic system by excluding statements that are not provable from your system, as meaningless.
The problem is not how to exclude the unprovable statements. The problem is that the unprovable statements will include statements that are true, but that the system can't prove. Some of those statements are probably not ones you care about, like "this statement is unprovable". But you have no way of knowing that all of the unprovable true statements are like that. Some of them might be ones you do care about.
Thus, the real import of Goedel's theorem is not "you need to exclude unprovable statements"; it is "the intuitively attractive ideal of having a system in which every true statement you care about can be proved is not achievable".
The problem is not how to exclude the unprovable statements. The problem is that the unprovable statements will include statements that are true, but that the system can't prove. Some of those statements are probably not ones you care about, like "this statement is unprovable". But you have no way of knowing that all of the unprovable true statements are like that. Some of them might be ones you do care about.
Thus, the real import of Goedel's theorem is not "you need to exclude unprovable statements"; it is "the intuitively attractive ideal of having a system in which every true statement you care about can be proved is not achievable".