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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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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 a binary alphabet, and the set of sequences of integers, which is a set of sequences with an infinite alphabet. Both have the same cardinality as $\reale$.} 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 reasonable definitions 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}.