[asayers]

The proof offered in the abstract demonstrates a simple link between consistency and the validity of proof by contradiction. It shows that if mathematics is consistent (ie. ⊢Φ and ⊢¬Φ is impossible) then mathematics is consistent.

This is NOT a self-proof - it is a meta-proof. Taking arithmetic as an example, a self-proof of consistency would be a derivation of the consistency sentence (ie. "I am consistent", or "¬◻(0=1)") from the arithmetic axioms. That is not what we have here.

The existence of a proof of the consistency of a theory does not put it at risk from Godel's 2nd - for that we require that a theory prove its own consistency.

As far as I can tell, Hewitt begins with a proof that consistency and the validity of proof by contradiction are equivalent, and then proceeds on the grounds that consistency is proven internally to the theory - which it is not.