Are you asserting that this would be a testable attribute of a putative omniscient god that could not be demonstrated by a simulated god? Or merely that it is an attribute that could not be simulated, even if I as an observer would have no way of telling the difference between the real thing and a sufficiently properly rigged demo?
Definitely the latter. Probably also the former, but constructing the test (or proof that such a test exists) requires math skills I don't have and so there is some chance that it isn't possible after all.
If you are polynomial time (BPP), then for any problem whose answer can be demonstrated to you by an interactive proof protocol with an untrusted logically omniscient prover (IP), that protocol can also be executed by a prover who is limited to PSPACE. (The set of problems that can be thus proved is also PSPACE).
Proof: for any given BPP verification algorithm (you), the prover's task is selecting a strategy that maximizes your chance of acceptance. Which is equivalent to choosing the optimal move in a game tree with polynomial depth (because of your limite...
A friend recently asked how strongly I believe that my deconversion from Christianity was not a mistake. Here's my response, and for those of you who are not Christians, I'm just wondering what numbers you would give:
"There is a part of me that wants to say the chance is far less than 1 percent. But when I consider what 1% must mean about my ability to follow complex arguments and base my judgement on the right premises, it seems absurd to say that.
Trying to honestly estimate the chance that I'm wrong about the Bible being generally reliable is a fascinating exercise... I know the number is low, but I'm not sure how low.
Today I would give myself a 1 in 20 chance of being wrong. If I were to consider the arguments of 20 other groups similar to Christian theologians, I would probably misunderstand them at least 1 time in 20. After talking with 20 groups that have a very different worldview, I might think they are all are mistaken, but once in a while, maybe 5% of the time, it would actually be me.
Wow, 5%!?! If I convert that into "There is a 5% probability that the God of the Bible exists and will send me to hell", I feel scared. But I know how to cheer myself up: I just say, "No way, the chance I'll end up in hell MUST be less than 5%. After all, the God of the Bible is CLEARLY just a big, mean alpha-monkey and... [rehearse all the atheistic arguments here]".
This back-and-forth from certainty to uncertainty makes me feel like I'm doing something seriously wrong.
So what about you? What chance do you place on some variant of Christianity turning up to be true, and what chance do you think a god of some sort exists?"
Numbers please.