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Note that the previous definition of prediction does not say that it is feasible to actually predict everything, it only means that prediction is possible for an all-knowing being. [TODO: say why this does not matter: requiring that prediction is actually possible could only make the paper stronger. However, I chosed to argue about a better result.] A related case is the following: It is possible that almost all macroscopic events can be predicted very precisely using quantum physics. Assuming that this is indeed the case, many of these predictions require too many computational resources, making them infeasible. I am requiring even less than this, I am allowing axiom systems where there is no way to infer a prediction from the axiom system, but if one checks all possible models of that system, the prediction turns out to be true.  Also, a mathematical model for a universe needs a set of measuring units. For each of the universes containing intelligent beings, let us take a fixed set of measuring units covering everything that those beings would measure. As an example, when measuring distance in our space we could use meters, light seconds or various other measuring units. The measuring unit is not important as long as we pick something.  We can define the \definitie{fraction of the world} that is modelled by an axiom system in at least three ways:  \begin{enumerate}  \item As the fraction of the observable space for which the axiom system predicts something with a reasonable error margin.  \item As the fraction of the optimal set of axioms that is implied by the current axiom system.  \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 space.  \end{enumerate}  We can use any In all  of these definitions (and many other reasonable ones) for cases,  the reminder predictions made only from artificial constraints imposed by this paper (the world can be modelled mathematically, contains intelligent beings) should not count towards the fraction  of the world that is modelled by an axiom set. In other words,  this paper. \definitie{fraction of the world} is actually the fraction of the world that is modelled without 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.  Then we would have three possible cases. First, those intelligent beings could, at some point in time, find an axiom system which gives the best predictions that they could have for their world, i.e. which predicts everything that they can observe. In other words, they wouldn't be able to find anything which is not modelled by their system. We could relax this "best axiom system" condition by only requiring an axiom system that is 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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\section{Approximations}  For each of Not being able to model  the universes containing intelligent beings, let us take a fixed set of measuring units covering everything that those beings would measure. As an example, when measuring distance in our space we could use meters, lights seconds or various other measuring units. The measuring unit entire universe precisely  is not important as long as that bad if we can at least have an approximation of most of the universe. Let us see how much  we pick something. can approximate.  Then Let us say that  \definitie{predicting things with a precision $\epsilon$} means that when predicting that something is of size $l$, then the actual size is in the range $(l(1-\epsilon), l(1+\epsilon))$. One could give a similar definition by using a statistical distribution that depends on the size $l$ and the precision $\epsilon$ instead of just using an interval. [TODO: Do I need this definition?] [TODO: I should use precision and accuracy as defined in the standard way e.g. here: https://en.wikipedia.org/wiki/Accuracy_and_precision]  Also, given Given  a time length $s$, we say that something \definitie{happens rarely} if in any given unit volume of space the time between two occurrences of that something is at least $s$. One could give similar definitions based on the probability of an intelligent being observing that something. [TODO: Say that we are using any reasonable definition for measuring things with a given precision at the beginning.]  Let us now do the following: 
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\item Fix $\delta \gt 0$ and say that we care about measuring things which are larger than $\delta$ [TODO: replace epsilon with delta where needed]. 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 $s$ and ignore things which happen rarely.  \end{itemize}  We could use any reasonable definition of "measuring" and "happen rarely". 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 $t\ge 0$, an acceptable error [TODO: use the right term here. I had a note to use precision/accuracy]  $\epsilon \ge 0$ and a probability $q\ge 0$ for our predictions which is the probability of a random prediction to be successful  [TODO: did I define this?] this? Should I move this at the end and say that this is the probability given the previous constraints?]  and let us denote by $f$ with $0 < f \le 1$ the fraction of the world where we can make predictions using about what happens after  the given time length $t$, with  the acceptable error $\epsilon$, $\epsilon$ and  having a probability $p$ $q$  that the prediction is correct. Then, if the world is not designed, we have a countable number of finite observable [TODO: is observable the right term?] descriptions out of a $\reale$ total number of 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 an error $\epsilon$, with a probability $p$ 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), prediction, we are just restating the present),  $\epsilon = \infty$ (which means that our predictions have no connection to the reality), $p=0$ (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. [TODO: Should I replace $f=0$ with "the minimal fraction absolutely needed", because having a space-time is a property of the entire universe, so f may not be zero? On the other hand, it does not allow any prediction. Should I add a footnote?] 
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[TODO: I should think about what happens when replacing $p$ with a distribution probability.]  Besides the "finite description for a non-zero fraction of the observable universe" property, we can look at some of the properties of our universe like homogeneity, isotropy or having the same forces acting through the entire universe. space or for all moments in time [TODO: Make sure that these are distinct].  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 any of these properties, sharing the same (mathematical space) 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. 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.  [TODO: Start rewriting from here.] put this below and link it to the conclusion.]  [TODO: We experience gravity differently at various times and places - tides, variation from one place to another on Earth, on the Moon, when falling, although the law that describes gravitation does not change. We could imagine an universe where the actual law changes.] 
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[TODO: Fix spaces between math mode and punctuation.]  [TODO: Fix the usage of I and we.]  [TODO: Decide when I use axiom set and when axiom system. Say explicitly that they mean the same thing.]  [TODO: Use can't, won't, isn't and can not, will not, is not consistently.] [TODO: The intelligent beings can have $n$ incompatible models that would predict everything.]  [TODO: Think a bit more about the fact that even statistically we can't model more than $0%$].  [TODO: How can we observe the universe without having uniforme laws?]