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Manfred comments on Results from MIRI's December workshop - Less Wrong
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I'm confused about what you want when you say this. I think an example or two would help me.
My best guess: you want the assigned probability to equal the "ideal" probability (as laid out in the probabilistic reflection paper) when we take the limit of computing power going to infinity. It's difficult to take this limit for Abram's original proposal, but if you make it doable by restricting it to S, now there can be things that are true but not provable using statements in S, so infinite computing power doesn't get us there.
It's actually not too hard to demonstrate things about the limit for Abram's original proposal, unless there's another one that's original-er than the one I'm thinking of. It limits to the distribution of outcomes of a certain incomputable random process which uses a halting oracle to tell when certain statements are contradictory.
You are correct that it doesn't converge to a limit of assigning 1 to true statements and 0 to false statements. This is of course impossible, so we don't have to accept it. But it seems like we should not have to accept divergence - believing something with high probability, then disbelieving with high probability, then believing again, etc. Or perhaps we should?