Suppose T is unprovable iff. necessarily, everyone (unrestrictedly) fails to prove it. Suppose there is (as I'm sure there is) some true theorem T of such complexity that no possible finite being could prove it. Suppose God does not prove it either, since of course he has no need of proofs. It is then non-trivial that no one has proved the unprovable, and yet non-trivially so, since among the unprovable things are necessarily true things. It's non-trivial that some necessarily true propositions are not provable.
Suppose T is unprovable iff. necessarily, everyone (unrestrictedly) fails to prove it. Suppose there is (as I'm sure there is) some true theorem T of such complexity that no possible finite being could prove it. Suppose God does not prove it either, since of course he has no need of proofs. It is then non-trivial that no one has proved the unprovable, and yet non-trivially so, since among the unprovable things are necessarily true things. It's non-trivial that some necessarily true propositions are not provable.
ReplyDeleteIt is then non-trivial that no one has proved the unprovable, and yet non-trivially so, since among the unprovable things are necessarily true things.
ReplyDeleteYikes, nice sentence. Let's try this,
No one has proved the unprovable, and yet non-trivially so, since among the unprovable things are necessarily true things