= Dan Asimov Well, that may well be how they "do it". But my conviction is that, regardless of how they try to do this, there is no such thing as predicting, in any sense, an *arbitrary* function f [...] in the sense of determining anything at all about f(x) for x >= t from the information given by f(x) for x < t .
Well they are claiming *some* "sense" that survived refereeing (despite notational horrors), which is why I'm hoping someone who likely understands these things better than I (eg you!) will read the actual article and provide some further explanation. Or at least think about the construction I described and what it might lead to? I suspect this is one of those "watch me pull a rabbit out of my hat" tricks where it turns out that it really isn't a rabbit nor is it actually a hat. But even so I'd like to at least better understand what kind of rodent haberdashery is going on here. The authors say a lot of peculiar but suggestive things, often one right after another: "at almost every instant the strategy predicts some [valid] ''epsilon-glimpse'' of the future" "these results do not give a practical means of predicting the future" "the mu-strategy will correctly predict the present from the past on a set of full measure" "One needs to be cautious about interpreting that the mu-strategy is correct with probability 1. For a fixed true scenario, if one randomly selects an instant t then [it] does tell us [it] will be correct at t with probability 1. However if one fixes the instant t, and randomly selects a true scenario, then the probability that [it] is correct might be zero, or might not even exist, depending..." "...there is a scenario v in which strategy g is wrong at every point in W..." "the mu-strategy still almost always correctly predicts some of the future" With regard to AC they say it's "necessary to prove the existence of such strategies, [since] ZFC is adequate but ZF is not" having (I think) just argued that the contrary implies "there exists a set of reals that does not have the property of Baire (and is not Lebesgue measurable)". Alas I have no idea what that concretely means, other than gathering that there's subtlety hidden in "an *arbitrary* function f". Help, anyone?