You’re confusing consistency with a proof of consistency.
Theorem 56: Consistency implies no proof of consistency.
Which is of course where you get:
Proof of consistency implies inconsistency.
Which gives you:
Proof of consistency implies anything.
I think you’re right.… I was commiting the same mistake is above, using the first derivability condition and assuming that Peano arithmetic could treat it as a statement in Peano arithmetic—which it isn’t.
You’re confusing consistency with a proof of consistency.
Theorem 56: Consistency implies no proof of consistency.
Which is of course where you get:
Proof of consistency implies inconsistency.
Which gives you:
Proof of consistency implies anything.
I think you’re right.… I was commiting the same mistake is above, using the first derivability condition and assuming that Peano arithmetic could treat it as a statement in Peano arithmetic—which it isn’t.