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Quill_McGee comments on An Introduction to Löb's Theorem in MIRI Research - Less Wrong
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"if I can prove that if a version of me with unbounded computational resources is consistent then this is good, do it"
In this formalism we generally assume infinite resources anyway. And even if this is not the case, consistent/inconsistent doesn't depend on resources, only on the axioms and rules for deduction. So this still doesn't let you increase in proof strength, although again it should help avoid losing it.
If we are already assuming infinite resources, then do we really need anything stronger than PA?
A formal system may be inconsistent, but a resource-bounded theorem prover working on it might never be able to prove any contradiction for a given resource bound. If you increase the resource bound, contradictions may become provable.