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\section{Disclaimer}  The \paper \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 \paper{}  is different from what I have read about both of these, maybe being complementary to the fine-tuning argument. Since it started as a mathematical \paper \paper{}  I may use \ghilimele{we} instead of \ghilimele{I} more often than I should, but you should consider it an invitation to work together in discovering some ideas. And if some of those ideas are wrong or unclear\footnote{Given my lack of experience with philosophy this is more probable than I would like.}, I welcome counterarguments and feedback\footnote{Authorea allows everyone to comment on the \paper. \paper{}.  I may switch to a different commenting system if it turns out that something better is needed.}. % TODO: Maybe add a summary. 
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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 the 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 of this \paper, \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 \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. \section{Modelling possible worlds} 
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Then, in the following, we will say that we can \definitie{predict} something ($S$) whenever we have a set of axioms for which $S$ is uniquely determined by the state of the world at a subset of the previous points in time\footnote{Will be extended to statistical predictions in the next paragraph.}. If we are interested in predicting the state of the world at a given point $P$ and time $t$, a good choice for this subset could be a full section through $P$'s past (e.g. a plane which intersects it's past cone), i.e. a subset that separates $P$'s past in two parts, one which is before the subset and one which is after the subset\footnote{This means that 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.  In a deterministic universe, if we know the laws of the universe and its full state at a given time, we could, in theory, fully predict any future state. But an universe does not have to be deterministic and, even if it is, the only reasonable model available at a certain time may be a statistical one. Then we will allow using a set of axioms which only gives a statistical distribution for the state of the universe given its past (I'll call this a \definitie{statistical axiom set}). For the purpose of this \paper \paper{}  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. Let us restrict the possible worlds term to the worlds where we can make predictions and let us use only sets of axioms that allow predictions. As defined above, for a given world, a good set of axioms is one which allows us to make all possible correct predictions for that world (statistical or not). Using only good sets of axioms solves the too-general problem since such a set would describe its possible worlds in a way that does not leave out important things. Still, there is nothing that prevents such a set from going into too much detail. 
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\item As the fraction of the optimal set of axioms that is implied by the current axiom set.  \item As something between the first two cases, where we use a weighted fraction of the optimal axiom set, each axiom having a weight proportional to the fraction of the world where it applies. As an example, let us say that we have an infinite set of axioms, and that for each point in space we can predict everything that happens using only three axioms (of course, two different points may need different axiom triplets). Let us also assume that there is a finite set of axioms $S$ such as each point in space has an axiom in $S$ among its three axioms. Then $S$ would model at least a third of the entire world.  \end{enumerate}  In all of these cases, predictions made only from the artificial constraints imposed by this \paper, \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 of this \paper. \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 \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. 
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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.   Note that the fraction of the world that can be modelled is (non-strictly) increasing, but is limited, so it converges at some value. Also, the prediction error (it's not important to define it precisely here) is (non-strictly) decreasing and is limited, so it converges\footnote{The prediction error can be different for different kinds of prediction and for different parts of the world. However, even then, it will still be decreasing and limited, so, in order to avoid unneeded complexity, I have assumed that there is only one. This part of the \paper \paper{}  could be rephrased to handle the multiple prediction errors case. Note that even when replacing an old theory with one that covers more but has a higher prediction error one could still use the old one when it works and only use the new one otherwise, keeping the prediction error non-increasing.}. 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. Their convergence shows that all improvements converge to zero so, after some point, one can't make any meaningful improvement. However, there is another meaning of \ghilimele{meaningful} for which there are some worlds where one can make meaningful improvements forever. I will count this as the third case, although it is actually a subcase of the second case above. Of course, it will still be true that, after some point, these improvements do not really grow the fraction of the world that is covered by the set and they do not decrease the prediction error.  
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\section{Description probabilities}  For the purpose of this \paper, \paper{},  let us denote by \definitie{finite property} of something any property of that something which can be written using a finite number of words. Of course, all the properties that we will ever use in speech and writing are finite. Since we will use only finite properties here, let us drop \ghilimele{finite} and call any of them simply \definitie{property}. These observable descriptions of possible worlds are general enough and different enough that it's hard to say something about them, except that they make sense in a mathematical way. Still, given any property $X$ we could try to see what is the chance that it's true in the set of observable descriptions. 
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\item Fix $\delta > 0$ and say that we care about measuring things which are larger than $\delta$. This means that we can have three sizes $a$, $b$ and $c$ with $a=b$ and $b=c$ but $a\not=c$. This should be fine as long as we're aware that equality here actually means that the difference is smaller than $\delta$.  \item Fix a time length $s$ for the happens rarely definition and ignore things which happen rarely.  \end{itemize}  We could actually use any reasonable definition of measuring and happens rarely, the ones above are provided as an example. Then we could say that the important things are the ones which are larger than $\delta$ and which do not happen rarely. Let us also fix an arbitrary time length $t > 0$, a relative error $\epsilon \ge 0$ and a probability $q > 0$ which is the probability of a random prediction to be successful given the previous constraints and let us denote by $f$ with $0 < f \le 1$ the fraction of the world\footnote{As above, everything that can be inferred from the artificial restrictions imposed by this \paper \paper{}  to the possible worlds is not considered a part of $f$.} where we can make predictions about what happens after the given time length $t$, with the relative error $\epsilon$ and having a probability $q$ that the prediction is correct\footnote{This could be replaced by \ghilimele{having a probability greater or equal to $q$ that the prediction is correct}, which would also work when having a richer probability distribution for the correctness of the prediction.}. These restrictions do not change the fact that we still have a countable number of finite observable descriptions out of a $\reale$ total number of descriptions and, if our world is not designed, we should use continuous probability distributions over these descriptions. Then, for any continuous distribution, the probability of having a finite description with which we can make predictions for a time length of $t$, with a relative error $\epsilon$, with a probability $q$ and for a fraction of the world $f$, is $0$. To have a non-zero probability we must be on the boundary of these restrictions, so either $t = 0$ (which means that we are not making any prediction, we are just restating the present), $\epsilon = \infty$ (which means that our predictions have no connection to the reality), $q=0$ (which means that our predictions always fail) or $f=0$. We can discard the first option since then we would have no predictions. We can also discard the second and the third since such a description would not be useful in any way. The only remaining option is that $f=0$; as argued in an example above, a description with $f=0$ can actually make sense. Therefore, with probability $1$, we have $f=0$ and the world has an infinite model.