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\end{enumerate}  In all of these cases, the predictions made only from artificial constraints imposed by this paper (the world can be modelled mathematically, contains intelligent beings) should not count towards the fraction of the world that is modelled by an axiom set. In other words, this \definitie{fraction of the world} is actually the fraction of the world that is modelled without what is absolutely needed because of the constraints imposed here.  We can use any of these definitions (and many other reasonable ones) for the reminder of this paper. Then we would have three possible cases. cases\footnote{All of these assume that the intelligent beings use a single axiom system for predicting. It could happen that they use multiple axiom systems which can't be merged into one. One could rewrite the paper to also handle this case, but it's easy to see that the finite/infinite distinction below would be the similar.}.  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, i.e. which predicts everything that they can observe. In other words, they wouldn't be able to find anything which is not modelled by their system. We could relax this \ghilimele{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. 
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[TODO: Fix the usage of I and we.]  [TODO: Decide when I use axiom set and when axiom system. Say explicitly that they mean the same thing.]  [TODO: Use can't, won't, isn't and can not, will not, is not consistently.]  [TODO: The intelligent beings can have $n$ incompatible models that would predict everything.]  [TODO: Think a bit more about the fact that even statistically we can't model more than $0$].  [TODO: How can we observe the universe without having uniforme laws?]  [TODO: define "mathematical prediction" or something like that and use it here, since I have two meanings of prediction - I think this is done/not needed, I added a footnote in the only place where I use the normal meaning of prediction. I have to check. Maybe I should be more explicit about this.]