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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, so probably many people have already thought about them. I tried to find this idea on the internet, but 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}}. 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 to actually develop it. 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.}. 
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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 (including models which need an infinitely long description, but which still follow our rules for reasoning).  This notion of \ghilimele{model} is not precise enough. Let us restrict the \ghilimele{possible worlds} term even more, to the possible worlds that we could model mathematically, even if that may leave out some worlds.This was done for simplicity, the constructions made below also work if we fix a non-zero number $f$ between $0$ and $1$ and require that a fraction of at least $f$ of the world can be modelled mathematically.  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.}. If nobody designed our world then we have no way of preferring a possible world over another, except that there is one of them in which we live. In other words, if our world is the only one that exists, then any other possible world is as likely to have existed as well. If multiple worlds exist, by picking a random world, ours has a the same chance of being picked as any other. 
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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 \ghilimele{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 (possibly infinite) paper. This is not a well defined notion, 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}. With an axiom set chosen in this way we would also solve the \ghilimele{too-specific problem} in the finite case. In the infinite case we [TODO: have?] 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}. 
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Then, for the given intelligent beings we would say in the first case that their universe has a \definitie{finite observable description} and in the second and third case that it has an \definitie{infinite observable description}. Of course, a possible universe $U$ could have multiple types of intelligent beings, each type perceiving the universe in a different way. Because of this difference in perception, for some intelligent beings the universe $U$ may have a finite observable description while for others it may have an infinite observable description.  If we can have infinite descriptions, then the set of optimal systems of axioms\footnote{Using \ghilimele{set of axioms} in some contexts may make the text harder to read, so I'm replacing it with \ghilimele{system of axioms}.} would have the same cardinality as the set of real numbers $\reale$. \footnote{Note that for easier writing, I will sometimes use $\reale$ $\reale$\  as a number, i.e. \ghilimele{the number of is $\reale$} instead of more complex phrases that uses \ghilimele{cardinality}.} Indeed, for the planets-on-a-line example above, we could select any subset of planets and get an universe with an optimal set of axioms that is distinct from any other subset. The set of all subsets of $\naturale$ has the same cardinality as $\reale$, so the set of optimal sets of axioms would have at least this cardinality. On the other hand, each set of axioms is written using an at most countable number of symbols over a finite alphabet, so there can't be more than $\reale$ sets of axioms.\footnote{See, e.g., \href{https://en.wikipedia.org/wiki/Cardinality_of_the_continuum}{https://en.wikipedia.org/wiki/Cardinality_of_the_continuum}. The cardinal of the set of letter sequences is between the power set of $\naturale$, which can be easily put into 1:1 correspondence with the set of letter sequences over axioms. [TODO: Add  a binary alphabet, and the set of sequences of integers, which is footnote or  a set of sequences chapter at the end  with an infinite alphabet. Both have more mathematical justification for this. Maybe add there  the same cardinality as $\reale$.} half-proof given here. Maybe I shouldn't bother.]  When talking about a mathematical description of the universe as one sees it, it is obvious that the description may depend both on time and place, i.e. the laws of the universe as observed at a given time and place can be quite different from the laws at another time and/or place. If these differences are unpredictable, then an intelligent being will never be able to find a full mathematical description of the universe, even if we assume that it could live through all these changes (as time passes, and/or as it moves through the space). 
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Let us now do the following:  \begin{itemize}   \item Restrict ourselves to universes in which we can measure things with real numbers.  \item Fix $\delta > \gt  0$ 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 \ghilimele{happens rarely} definition and ignore things which happen rarely.  \end{itemize}  We could actually use many any  reasonable definitions 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 $t\ge 0$, a relative error $\epsilon \ge 0$ and a probability $q\ge 0$ 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}. Remember that Then, if the world is not designed,  we have a countable number of finite (observable) descriptions out of a $\reale$ total number of descriptions. Then,if the world is not designed,  for any continuous distribution 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 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 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, 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 given a set of universes having these properties, sharing the same mathematical space (e.g. $\reale^3$) 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. 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}  The strongest conclusion of this argument is that, from the hypothesis that the universe is not created and a few basic mathematical properties, properties  one can predict, with $100\%$ certainty, that we can't know how a non-zero fraction of the observable part of our universe works, for many reasonable definitions of \ghilimele{fraction}. Either we can't apply any scientific theory to the distant past, future, or to distant places (e.g. most of astronomy would be become  just a joke), and we will never be able to do that, or one of the starting axioms must be false. I'm not betting on either astronomy being a joke or the mathematical statements being false. In other words, if our world is not designed, there is a good chance that we may know a lot about what happens on Earth, maybe something about what happens in our solar system, we almost surely don't know what happens in our galaxy and outside of it and we will never know a non-trivial part of what we can observe. Also, we have a good chance of knowing how the world works now and in the near past and future, but we probably don't know what were the physical laws in the distant past or how they will be in the distant future.