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\section{Disclaimer} r\section{Disclaimer}  The \paper{} below started as a mathematical attempt to understand what it would mean to live in a world that is not designed, but, in the end, the mathematical part turned out to be rather small, containing only a few simple properties about set cardinalities and probabilities. I think that the non-mathematical ideas are fairly obvious consequences of the mathematical ones, so many people have already thought about them – I have also found quotes from various people that seem to hint at the idea below. However, I did not manage yet to find anyone drawing the same conclusions in the same way. The closest I could get is the idea that the order of the Universe implies or suggests that there is a God. The fine-tuning of the Universe is also close\footnote{\href{https://en.wikipedia.org/wiki/Fine-tuned_Universe}{https://en.wikipedia.org/wiki/Fine-tuned\_Universe}}. However, I think that what I'm presenting in this \paper{} is different from what I have read about both of these, maybe being complementary to the fine-tuning argument. 
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Many people believe that the world is designed and created and that it's unreasonable to believe that any world can exist without being created, and I agree with them. However, these beliefs are not shared by everyone, so it's worth thinking about what this means. If our world is created, then it's likely to be the way it is because its Creator\footnote{Not everybody that believes that the world is created thinks that God created it. Still, I hope that they would agree that capitalizing the Creator of this world is reasonable.} wanted it to have certain properties. In order to understand why our world works the way it does, one would need to understand the intent of its Creator. While that is interesting in itself, I will not try to pursue it here, except for a small related paragraph at the end.  For the reminder remainder  of this \paper{}, let us consider the other case and assume that our world was not designed and created. If that's true then there may be other worlds\footnote{We don't have any proof for the existence of other worlds, but one could expect them to exist for the same reason that ours exists. If ours has no reason at all for existing, which is likely if it is not created, then it's likely that other worlds would also not need any reason for existing and would simply be. However, for the purpose of this \paper{} it does not matter if there are other worlds or not and maybe we will never be able to tell if other worlds exist or not.}. Even if there are no other worlds, one could easily imagine that ours worked in a different way, say that the speed of light is different or gravity works differently. We will denote by \definitie{possible worlds} these other worlds that either are or could have been. 
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\end{enumerate}  In all of these cases, predictions made only from the artificial constraints imposed by this \paper{}, e.g. that the world can be modelled mathematically or that it 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 above 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 remainder  of this \paper{}. Then we would have three possible cases\footnote{All of these assume that the intelligent beings use a single axiom set for predicting. It could happen that they use multiple axiom sets which can't be merged into one. One could rewrite this \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 set which gives the best predictions that they could have for their world, i.e. which predicts everything that they can observe and they wouldn't be able to find anything which is not modelled by their axiom set. We could also include here axiom sets that are 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 in a meaningful way. This could be the case if, e.g., they can model a part of their world, but modelling any part of the reminder remainder  would require adding an infinite set of axioms and no finite set of axioms would improve the model. In order to make the first two cases more clear, let us assume that those intelligent beings would study their universe and would try to improve their axiom sets in some essential way forever. Since they have infinite time available to them, they could use strategies like generating possible theories in order (using the previously defined order, which works for finite axiom sets), 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.