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Then we can say that prediction would mean predicting the state of the world (maybe at a given point) from the state of the world at a subset of the previous points in time. If we are interested in predicting the state at a given point $P$, this subset should include a full section through $P$'s past (e.g. a plane which intersects it's past cone), i.e. it should separate $P$'s past in two parts, one which is "before the subset" and one which is "after the subset"; all lines which fully lie in $P$'s past and connect a point which is before the subset with a point which is after the subset must go through the subset. One could think of similar definitions for predicting the entire state of the world. If needed, this definition could be changed to work for more concepts of space and time.  Note that in In  a non-deterministic world deterministic universe, knowing the laws of the universe and its full state  we may not always be able to could, in theory,  fully predict $P$'s state. In But an universe does not have to be deterministic and, even if it is, one could have only a statistical model for it. Then a set of axioms which only allows statistical predictions (I'll call  this case, a \definitie{statistical axiom set}) is fine and for the purpose of this document we don't need to make a difference between a non-deterministic universe and a deterministic one but for which we only have a statistical model. [TODO: add the probability to  the term "prediction" would mean reasoning about knowing a $0$ fraction of the universe.]  Also note that there are cases when one can't have  a statistical prediction. axiom set, e.g. when the perceived laws of the universe change in fully unpredictable ways from day to day (of course, this can happen without any change in the actual axiom set for the universe). [TODO: move this near the discussion about finite models].  If we define "prediction" in some useful way, as suggested above, and restrict the "possible worlds" term to the ones where we can make predictions, then it makes sense to use only systems of axioms that allow predictions. We will always use the best possible system for predictions (statistical or not). This solves the "too-general problem" since such a system would describe its possible worlds in a way that does not leave out important things. Still, there is nothing that prevents such a system from going into too much details. 
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[TODO: Start rewriting from here.]  And there is a second distinction that we should make. Even if the universe is deterministic, one could have only a statistical model for it. [TODO: add the probability to the above reasoning.] In a deterministic universe, knowing the laws of the universe and its full state we could, in theory, fully predict its future. But an universe does not have to be deterministic. In this case, we would search for the best set of laws for predicting the future state. Note that we could have an universe that seems non-deterministic for any finite set of laws but which has an infinite set of laws under which it is deterministic. Indeed, it seems that in a case in which we can easily observe the effects of some laws of the universe, we could probably infer a finite statistical law about it. In this case, the best set of laws would be the infinite one. [TODO: Think about finite statistics. Is it always true? Probably not, if the stats made in a day are completely different from stats made in another day. How frequent would it be? What does it mean?]  [TODO: Give examples in which our main assumptions about the universe, i.e. homogeneity and isotropy, are broken. Are these finite properties, or zero-probability ones? They are not finite, but considering that we can combine any at most countable set of homogenous and isotropic universes with compatible times into another universe, then it's likely that they are zero-probability ones. We need intelligent beings to be able to live through these changes, but even then it looks like we can combine a lot of universes into one, suggesting that these properties are zero-probability for many reasonable probability distributions. TODO: Give examples on how to combine. Say in a clear way what do I mean by combining a lot of universes into one, making it obvious why the probability should be zero. We experience gravity differently at various times and places - tides, variation from one place to another on Earth, on the Moon, when falling, although the law that describes gravitation does not change. We could imagine an universe where the actual law changes.]  When talking about a mathematical description of the universe as one sees it, it is obvious that the description may depend both on time and place of the observers (assuming that the universe has a concept of place that is close enough to ours). The laws of the universe as observed at a given time and place can be quite different from the laws that one can observe at another time and/or place. If these differences are unpredictable, then an intelligent being will never be able to find a full mathematical description of the universe, even if we assume that it could live through all these changes (as time passes, and/or as it moves through the space). Note that these beings must be able to live through the changes, otherwise the universe does not count for our problem.