I haven’t seen good attempts to answer that, just agitation about the problem, which is sad because it seems important. In my amateur syncretic speculations I try to look at theology from the lens of theoretical computer science (esp. algorithmic information theory) and there you have an infinite hierarchy of oracles, there’s no escaping diagonalization. It makes me wonder if human intuitions about omniscience &c. are screwed up because simple self-reference problems show some of our naive conceptions of infinity to be logically impossible. It’s possibly possible that a very clever, very fundamental formalization of the self vs. non-self (same vs. not-same) distinction would “solve” the problems but I don’t know if any philosophically-inclined mathematical logicians think that’s plausible.
There are also sideways-bending ideas about the role of “faith” in hypercomputation, and the possibility of logical (“acausal”) influence between arbitrarily distant oracle machines in the arithmetical hierarchy. (I’m not comfortable with the math, I can’t tell whether a machine’s oracle would “screen off” all higher degree oracles; I vaguely suspect the fomal analytical reifications are too brittle to say, but I’m totally not a mathematician.)
I sometimes try to analogize the self-reference/infinity problem to the Myerson-Satterthwaite theorem in mechanism design, where a seemingly simple epistemic problem turns out to have no solution. I find it funny to think of what the Myerson-Satterthwaite theorem and things like it would imply about a God that is actually three distinct persons.
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.
I haven’t seen good attempts to answer that, just agitation about the problem, which is sad because it seems important. In my amateur syncretic speculations I try to look at theology from the lens of theoretical computer science (esp. algorithmic information theory) and there you have an infinite hierarchy of oracles, there’s no escaping diagonalization. It makes me wonder if human intuitions about omniscience &c. are screwed up because simple self-reference problems show some of our naive conceptions of infinity to be logically impossible. It’s possibly possible that a very clever, very fundamental formalization of the self vs. non-self (same vs. not-same) distinction would “solve” the problems but I don’t know if any philosophically-inclined mathematical logicians think that’s plausible.
There are also sideways-bending ideas about the role of “faith” in hypercomputation, and the possibility of logical (“acausal”) influence between arbitrarily distant oracle machines in the arithmetical hierarchy. (I’m not comfortable with the math, I can’t tell whether a machine’s oracle would “screen off” all higher degree oracles; I vaguely suspect the fomal analytical reifications are too brittle to say, but I’m totally not a mathematician.)
I sometimes try to analogize the self-reference/infinity problem to the Myerson-Satterthwaite theorem in mechanism design, where a seemingly simple epistemic problem turns out to have no solution. I find it funny to think of what the Myerson-Satterthwaite theorem and things like it would imply about a God that is actually three distinct persons.
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.