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I Wanna Get Rid of God, Right?
David Lewis said that physicists are interested in describing all the facts that will hold throughout all its base in time. What you want is an axiom system which will best balance informativeness, giving us as much information as one can about all those facts in a simple way as possible. And something you mentioned earlier fits into this very well, because dividing the facts up into initial conditions and general dynamics turns out to be a way of doing that for a world like ours. Didn't have to be, this is a contingent feature in nature of our world, but it seems to be so for our world.