It is not that these statements are “not generally valid”, but that they are not included within the axiom system used by H. If we attempt to include them, there will be a new statement of the same kind which is not included.
Obviously such statements will be true if H’s axiom system is true, and in that sense they are always valid.
It is not that these statements are “not generally valid”
The intended meaning of valid in my post is “valid step in a proof” in the given formal system. I reworded the offending section.
Obviously such statements will be true if H’s axiom system is true, and in that sense they are always valid.
Yes, and one also has to be careful with the use of the word “true”. There are models in which the axioms are true, but which contain counterexamples to Provable(#φ) → φ.
It is not that these statements are “not generally valid”, but that they are not included within the axiom system used by H. If we attempt to include them, there will be a new statement of the same kind which is not included.
Obviously such statements will be true if H’s axiom system is true, and in that sense they are always valid.
The intended meaning of valid in my post is “valid step in a proof” in the given formal system. I reworded the offending section.
Yes, and one also has to be careful with the use of the word “true”. There are models in which the axioms are true, but which contain counterexamples to
Provable(#φ) → φ.