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First, let us note that having intelligent beings in an universe likely means that their intelligence is needed to allow them to live in that universe, which likely means that they can have a partial model of the universe. That model does not have to be precise (it could be made of simple rules like "If I pick fruits then I can eat them. If I eat them then I live.") and it can cover only a small part of their world, but it predicts something. Of course, their predictions do not have to be deterministic. Also, they might not be able to perceive the entire universe.  In the following we are interested only in how good an axiom set is for describing a given world, so when saying that someone predicts something using an axiom set this will not mean that a human-like intelligence is able to make that prediction, it means that in theory there is a mathematical way to make that prediction [TODO: nu suna bine.].  Then we would have three possible cases. First, those intelligent beings could, at some point in time, find an axiom system which gives the best predictions that they could have for their world, which means that they would stop searching for axiom systems, since what they have predicts everything that happens in their world and they can't find anything which is not modelled by their system. We could relax this "best axiom system" condition by only requiring an axiom system that is good enough for all practical purposes. As an example, for an universe based on real numbers, knowing the axioms precisely with the exception of some constants and measuring all constants with a billion digits precision might (or might not) be good enough. Only caring about things which occur frequently enough (e.g. more than once in a million years) could also be "good enough".  Second, those intelligent beings could reach a point where their theory clearly does not fully model the world, but it's also impossible to improve. This could be the case if, e.g., they can model a part of their world, but modelling any part of the reminder would require adding an infinite set of axioms and no finite set of axioms would get one a better model.  [TODO: Maybe say that "predict" means "predict given enough time and other resources" or something similar.]  Let us assume that if those intelligent beings could study their universe forever, they would try to improve their models in some essential way forever. Since they have an infinite time, they could use strategies like generating possible theories in order, checking if they seem to make sense and testing their predictions against their world, so let us assume that if there is a possible improvement to their current theory, they will find it at some point. Note that in this case the fraction of the world that can be modelled [TODO: Define separately] (if that notion makes sense) is increasing, but is limited, so it converges at some value. Also, the prediction error [TODO: define separately] is decreasing and is limited, so it converges. If the fraction converges at $1$ and the prediction error converges at $0$, then we are in the first case, because we reach a point when the fraction is so close to $1$ and the error is so close to $0$ that one would find them "good enough". If the fraction or the error converges to different values then we are in the second case.  There is also a third case when there is no reasonable way to define the fraction of the world that can be modelled, except when the fraction is $0$. As an example, imagine a world with an infinite number of earth-like planets that lie on one line and with humans living on the first one. The planets would be close enough and would have enough resources like food and fuel so that humans would have no issues travelling between them. Light would have to come to them in a different way than in our world and something else, not gravitation would keep them in place. The laws of this hypothetical world, as observed by humans, would be both close enough to the laws in our world so that humans can live on any of the planets, but also different in an easily observable way. Let us say that, starting at $10$ meters above ground, gravity would be described with a different function on each planet. On some planets it would follow the inverse of a planet-specific polynomial function of the distance, on others it would follow the inverse of an exponential function, on others it would behave in some way if the distance to the center of the planet in meters is even and in another way if the distance is odd, and so on. 
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There is a distinction that we should make. When predicting (say) weather we can't make long-term precise predictions, and this happens because weather is chaotic, that is, a small difference in the start state can create large differences over time. This could happen even if the universe is deterministic and we know the laws of the universe perfectly, as long as we don't know the full current state of the universe. However, as argued above, with probability $1$, our hypothetical intelligent beings would not be able to make predictions for a significant part of the universe because they would have no idea about how their universe works, not because they don't know its state precisely enough.  [TODO: I should think about what happens when replacing $p$ with a distribution probability.]  Besides the "finite description for a non-zero fraction of the observable universe" property, we can look at some of the properties of our universe like homogeneity, isotropy or having the same forces acting through the entire universe. It is harder to give a mathematical proof that these are zero-probability ones, but if we think that given a set of universes having any of these properties, sharing the same (mathematical space) and having at least two distinct elements, one can slice and recombine them in infinite ways, it is likely that these properties are also zero-probability ones. An example of such a combined possible universe is the one with infinite planets on a line mentioned above. In other words, the cosmological principle is (very) likely to be a zero-probability property.  [TODO: Start rewriting from here.]