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\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, soprobably  many people have already thought about them. them –  I tried have also found quotes from various people that seem  to find this hint at the  idea on below. However, I did not manage yet to find anyone drawing  the internet, but 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. I have found quotes from various people that seem to hint at the idea below, but I did not find yet anyone that tried these, maybe being complementary  to actually develop it. fine-tuning.  Since it started as a mathematical 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 document. I may switch to a different commenting system if it turns out that something better is needed.}.  % TODO: Maybe add a summary.  \section{Introduction}  This is an attempt to reason about why our world is the way it is and what we can reasonably believe about it.  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. here, except for a small related paragraph at the end.  For the reminder of this article, let us consider the other case and assume that our world was not designed and created.  For the reminder of this article, 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 this paper it does not matter if there are other worlds or not and probably maybe  we wouldn't 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}  First, let us note that there can't be any causal interaction between two different possible worlds. If two worlds are interacting, it's more reasonable to say that they are actually a single possible world with two parts.  How would a possible world look like? It could have exactly the same fundamental laws as ours, but withthe  matter organized differently. It could have similar laws, but with different universal constants. It could have different fundamental particles (or whatever the basic building blocks of our universe are, assuming that there are any). Or it could be completely different, i.e. different in all possible ways. It could be that our logic and reasoning are universal instruments, but it could also be that some of these possible worlds could be beyond what our reasoning can grasp and others could have properties for which our logic is flawed. Aknowledging that, let us see if we can say anything about the possible worlds that we could understand and could model in some way. In the following, the \definitie{possible worlds} term will denote only the possible worlds which we could model. But the \ghilimele{we could model} term here should not mean that we estimate our maximum capacity as humans in this world  to build a model in this universe, model,  since that is an arbitrary limit, we should also allow modelsof infinite size  that still follow our rules for reasoning. reasoning, but for which we would need an infinite amount of resources to build and use.  This notion of model is not precise enough. Let us restrict the possible worlds term even more, to the possible worlds that we could model mathematically, even if that may leave out some worlds. We will also do the reverse and say that all the models of any set of mathematical axioms which is at most countable\footnote{We could also go beyond countable axiom sets, but that would complicate things without any benefit.} and has at least a model are possible worlds\footnote{Even if some of those models seem outlandish, there could be something that follows those rules and that is completely separated from anything else, not interacting with any other universe in any way. I would say that this something would be an universe. If you don't like it, in the following I am going to restrict what I call a possible universe.}. 
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This ignores some important issues like the fact that it's reasonable to have a concept of time without having a well defined concept of the state of the world at a given time, so we could rephrase the definition above to include many other reasonable notions of space and time, e.g. we can include worlds where point is a concept and we can know which (point, time) pairs are before a given (point, time) pair.  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}. 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}. 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 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.   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.  Let us chose any formalism for specifying axioms that uses a finite alphabet and let us choose an arbitrary order relation on that alphabet. Also, let us consider only axiom sets which, when written on an infinite paper, use at most a countable number of symbols. Then, for each possible world, we could say that the best set of axioms is the smallest good one, smallest being defined as having the smallest length when written on the infinite paper. This is not a fully defined order relation, so let us also say that for sets of axioms having finite length we will break any ties by using the lexicographic order. For worlds having only sets of axioms of infinite length we can't use the same trick to break ties, but they all have the same length (countably infinite) so we will choose any such set of axioms and say that it's the best one\footnote{Note that we could try a few things to find a canonical set for the infinite case, e.g. whenever a set of axioms is implied by a smaller finite set we could replace it, but it's not clear that any such method will be enough}. enough.}.  With an axiom set chosen in this way we would also solve the too-specific problem in the finite case. In the infinite case we have made an arbitrary decision, but then the cardinality of the axiom set matches the complexity of the world anyway, even if there is some redundancy. If $U$ is an universe and $A$ is the axiom set chosen as above, then we would say than $A$ is the \definitie{optimal set of axioms for $U$}. If $A$ is a set of axioms which is optimal for some universe $U$ then we say that $A$ is an \definitie{optimal set of axioms}.  Now let us see if we actually need infinite length sets. We can have infinite sets\footnote{Note that for a given finite alphabet, i.e. in the current context, the infinite-length-set notion is identical to infinite-set.} of axioms, and there is no good reason to reject such sets and to ignore their possible worlds, so we will take them into account. It is less clear that we can't replace these infinite sets with finite ones. Indeed, let us use anybinary  encoding allowing us to represent these sets as binary strings, i.e. strings over a finite alphabet, e.g.  as binary functions over the set of natural numbers (it's not important to use binary strings, the construction below would work with many other mathematical objects). The For the given example, the  encoding of a set of axioms $A$, $encoding(A)$, would be a function from the natural numbers to a binary set, giving the value of the bit for each position in the encoding, $encoding(A):\naturale\longrightarrow \multime{0, 1}$. Then the following scenario becomes plausible: possible:  for any universe $U$ with an infinite set of axioms $A$, we can consider $U+encoding(A)$ to be an universe in itself which has $encoding(A)$ as part of its state at any moment in time. Then it's likely that we can find a finite set of axioms which allows predictions for such an universe. While, strictly speaking, this would be a different universe than the one we had at the beginning, it may seem similar enough to it so one may be tempted to use only finite sets of axioms. On the other hand, using only finite sets of axioms in this way seems to be some sort of cheating. In order to get a more honest system of axioms, we should define the state of the universe as what could be changed from inside. As an alternative, we could work with very specific sets of axioms, e.g. we could only talk about worlds which have $\reale^4$ as their space and time, whose objects are things similar to the wave functions used by quantum mechanics and so on.  \section {Modelling from inside}  However, Many of these possible worlds would  not all universes are interesting, so let be able to have intelligent life inside of them, which makes them less relevant. Let  us focus only on worlds the ones  which could contain intelligent beings and let us think about how the intelligent beings would model their world. Even more, let us include only world worlds  where there intelligent beings would be similar enough to us in that they can use logic and mathematics, but they wouldn't be able to process an infinite amount of that in a finite time. Let us note that in worlds with an infinite optimal set of axioms these intelligent beings they will never have a complete description of how their world works, but they will may  be able to build better and better models. Let us assume that those intelligent beings are continuously trying to find better models for their world and that they are reasonably efficient at this.  As a parenthesis, note that until now we restricted the possible world concept several times. The argument below also works with larger possible world concepts as long as those worlds have a few basic properties (e.g. one can make predictions and it can contain intelligent beings) and and,  at the same time time,  it is plausible that our world is such a possible world. 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, e.g. it could be made of simple rules like \ghilimele{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 should predict\footnote{This is the only place where predict means that the beings can actually say something about the future instead of a theoretical way of making predictions. Everywhere else we're using the previous definition of prediction which only requires that prediction is possible for a being which can take full snapshots of the universe and can go through all the possible models of an axiom set.} something. Of course, these predictions do not have to be deterministic. Also, they might not be able to perceive the entire universe. 
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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 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}. 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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\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 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}. prediction.}.  Then, if the world is These restrictions do  not designed, change the fact that  we still  have a countable number of finite (observable) 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. 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.  Besides the \ghilimele{finite description for a non-zero fraction of the observable universe} property, we can look at some of the properties of our universe like having the same forces acting through the entire space, for all moments in time. It is harder to give a mathematical proof that these are zero-probability ones, but if we think that but,  given a that any finite  set of universes having these properties, sharing the same mathematical space (e.g. $\reale^3$) and having at least two distinct elements, one objects inside them,  can slice be sliced  and recombine them recombined  in infinite ways, producing infinitely many additional universes,  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. Similarly, if we take the rules for how the universe works as we perceive them,most likely  there is a zero chance that they would apply through the entire universe and a very low chance that they would apply outside of earth / our solar system. \section{Conclusion} 
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[TODO: Put an e-mail address in the footnote in the introduction.]  [TODO: Things to check at the end: my macro is used for \ghilimele{quotes}.]  [TODO: Decide between article, document and paper.]  [TODO: Where should I use universe and where should I use world?]  [TODO: Is the paragraph start spacing reasonable?]