You made a good point, so I inverted it. I think I agree with your statements in this thread completely. (So far, absent any future change.) My prior comment was not intended to indicate an error in your statements. (So far, in this thread.)
If there is a way I could make this more clear in the future, suggestions would be appreciated.
Elaborating on my prior comment via interpretation, so that it’s meaning is clear, if more specified*:
[A] it’s a contradiction to have a provable statement that is unprovable, [B] but it’s not a contradiction for it to be provable that a statement is unprovable.
[A’] It’s a contradiction to have an unprovable statement that is provable, [B’] but it’s not a contradiction for it to be unprovable that a statement is provable.
A’ is the same as A because:
it’s a contradiction for a statement to be both provable and unprovable.
While B is true, B’ seems false (unless I’m missing something). But in a different sense B’ could be true. What does it mean for something to be provable? It means that ‘it can be proved’. This gives two definitions:
a proof of X “exists”
it is possible to make a proof of X
Perhaps a proof may ‘exist’ such that it cannot exist (in this universe). That as a consequence of its length, and complexity, and bounds implied by the ‘laws of physics’* on what can be represented, constructing this proof is impossible. In this sense, X may be true, but if no proof of X may exist in this universe, then:
Something may have the property that it is “provable”, but impossible to prove (in this universe).**
*Other interpretations may exist, and as I am not aware of them, I think they’d be interesting.
You made a good point, so I inverted it. I think I agree with your statements in this thread completely. (So far, absent any future change.) My prior comment was not intended to indicate an error in your statements. (So far, in this thread.)
If there is a way I could make this more clear in the future, suggestions would be appreciated.
Elaborating on my prior comment via interpretation, so that it’s meaning is clear, if more specified*:
A’ is the same as A because:
While B is true, B’ seems false (unless I’m missing something). But in a different sense B’ could be true. What does it mean for something to be provable? It means that ‘it can be proved’. This gives two definitions:
a proof of X “exists”
it is possible to make a proof of X
Perhaps a proof may ‘exist’ such that it cannot exist (in this universe). That as a consequence of its length, and complexity, and bounds implied by the ‘laws of physics’* on what can be represented, constructing this proof is impossible. In this sense, X may be true, but if no proof of X may exist in this universe, then:
Something may have the property that it is “provable”, but impossible to prove (in this universe).**
*Other interpretations may exist, and as I am not aware of them, I think they’d be interesting.
**This is a conjecture.
Thanks for clarifying