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 ...
I am about to graduate from one of the only universities in the world that has a high concentration of high-caliber analytic philosophers who are theists. (Specifically, the University of Notre Dame, IN) So as not to miss this once-in-a-lifetime opportunity, I have sent out emails asking many of them if they would like to meet and discuss their theism with me. Several of them have responded already in the affirmative; fingers crossed for the rest. I'm really looking forward to this because these people are really smart, and have spent a lot of time thinking about this, so I expect them to have interesting and insightful things to say.
Do you have suggestions for questions I could ask them? My main question will of course be "Why do you believe in God?" and variants thereof, but it would be nice if I could say e.g. "How do you avoid the problem of X which is a major argument against theism?"
Questions I've already thought of:
1-Why do you believe in God?
2-What are the main arguments in favor of theism, in your opinion?
3-What about the problem of evil? What about objective morality: how do you make sense of it, and if you don't, then how do you justify God?
4-What about divine hiddenness? Why doesn't God make himself more easily known to us? For example, he could regularly send angels to deliver philosophical proofs on stone tablets to doubters.
5-How do you explain God's necessary existence? What about the "problem of many Gods," i.e. why can't people say the same thing about a slightly different version of God?
6-In what sense is God the fundamental entity, the uncaused cause, etc.? How do you square this with God's seeming complexity? (he is intelligent, after all) If minds are in fact simple, then how is that supposed to work?
I welcome more articulate reformulations of the above, as well as completely new ideas.