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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 $\reale$\ 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. [TODO: Add a footnote or axioms.\footnote{See e.g. \href{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 chapter at binary alphabet, and  the end set of sequences of integers, which is a set of sequences  with more mathematical justification for this. Maybe add there an infinite alphabet. Both have  the half-proof given here. Maybe I shouldn't bother.] 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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\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 any many  reasonable definition 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}. Then, if the world is not designed, Remember that  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 become be  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.