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Will_Sawin comments on The Cartoon Guide to Löb's Theorem - Less Wrong
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Yes, but you can get from "there is no proof of proposition C" to "there is no proof of the canonical false proposition F". The second is often taken to be the definition of the provability of Peano's consistency, I believe.
It think the provability of consistency would be:
"There is a proof that there is no proof of the canonical false proposition F"
I'm using Peter Smith's definition (see http://www.logicmatters.net/resources/pdfs/gwt/GWT.pdf , Godel without too many tears).
In definition 59 (on page 1 of part 10), he identifies consistency with "not a proof of the canonical false statement". This is a valid statement within Peano arithmetic. And it's logical consequences are... anything.
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.