Yeah, I am; it really depends on how you define "true". I prefer this to be interpreted as "true in all models" so sentences are "true" when they are tautological consequences of a theory.
Using this definition, all "true" sentences are provable in first-order logic.
The (usual) Gödel sentence is true in the intended model of arithmetic, but I don't really like this property being referred to as "truth" without qualification.
> "true but unprovable" within a given system
Not sure about this: I don't think you can really say something is "true in a system" if it isn't provable. You can only assert its truth by saying it's "true in the intended model" without qualification, or by doing some meta-reasoning in a more powerful system outside the original one.
Yeah, I am; it really depends on how you define "true". I prefer this to be interpreted as "true in all models" so sentences are "true" when they are tautological consequences of a theory.
Using this definition, all "true" sentences are provable in first-order logic.
The (usual) Gödel sentence is true in the intended model of arithmetic, but I don't really like this property being referred to as "truth" without qualification.
> "true but unprovable" within a given system
Not sure about this: I don't think you can really say something is "true in a system" if it isn't provable. You can only assert its truth by saying it's "true in the intended model" without qualification, or by doing some meta-reasoning in a more powerful system outside the original one.