| > Consistent axiomatic systems require a Gödel sentence that is unprovable in the system; it requires a logical tautology or else your system will contain an infinite regress. I don't see the connection between these two statements.
Yes, in a consistent system with a finite description, a statement in that system which can be interpreted as "the system [description of the system] cannot prove [this statement]" cannot be proven (also, where "can be interpreted as" is taken to mean the sensible thing that it should mean) (though, sufficiently weak systems can have languages incapable of expressing such a statement) . But the second part of that sentence, after the semicolon, appears to be talking about something else? It seems like it is describing the Münchhausen trilemma , except applied to mathematical proofs, and giving an argument that a system needs axioms (and/or rules of inference).
While a system of proof does need axioms and rules of inference in order to conclude anything, this is not because of the incompleteness theorem. The two do not seem particularly connected to me. So, I don't see why you connected the two statements with a semicolon. Perhaps I misunderstood what you meant by the second statement in the sentence? Also, when you say "logical tautology", are you talking about purely logical tautologies, or are you talking about axioms? The way you are using it makes it seem like you are talking about axioms, which I wouldn't call logical tautologies. But maybe you do mean what I would mean if I was to refer to "logical tautologies"? In that case, uh, I don't think logical tautologies are all that much of a foundation? |