There are also sideways-bending ideas about the role of "faith" in hypercomputation,

I'm slightly more familiar with the theory of infinite cardinals than hypercomputation. Well, inaccessible cardinals and large cardinal axioms more generally have the property that their consistency can't be proved in ZFC in a very strong sense, i.e., adding any number of Godel statements doesn't help. Conversely, they can prove the consistency of ZFC unconditionally.

More generally, there is a hierarchy of large cardinal axioms where each one unconditionally implies the consistency of the ones below it but by Godel's second incompleteness theorem, they're consistency can't be proven (in a strong sense) from any ones below it.