'Given huge sets, much like reality, almost anything can be "proven" at some sort of local scale.'
I don't think you're right, I think that some things aren't tractable to this kind of analysis. Obv. Godel type things, but if you believe in freedom of will and an open universe then a lot of other things. For example, the path of true love, the next line of a poem, the summer after next's hot fashion trend.
More importantly (less) international events, earthquakes, solar flares.
Given enough data, while you can't predict exact actions, you can predict general trends with good accuracy. For example, given your complete post history and metadata about those posts and your views of HN, someone could fairly easily predict what subjects you will upvote, what subjects you'll comment on (and a good idea of the tone of your comment), and so forth.
We're creatures of habit, and once you have enough information to identify those habits, you can do a pretty good job at predicting what we will do. Heck, with the little data I have access to, I can be pretty sure which articles I will see comments from big posters like jacquesm and tptacek, and what their comments will contain. I can't predict every story they will comment upon, nor the exact details of the comments, but at a higher level it is definitely predictable.
I don't think you're right, I think that some things aren't tractable to this kind of analysis. Obv. Godel type things, but if you believe in freedom of will and an open universe then a lot of other things. For example, the path of true love, the next line of a poem, the summer after next's hot fashion trend.
More importantly (less) international events, earthquakes, solar flares.
Fractals, chaos and incompletness.