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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 naming capitalizing  the creator Creator  of this worldas \ghilimele{the Creator}  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. Let us consider the other case: for most of this article, let us 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 we wouldn't be able to tell if other worlds exist or not.}. Even if there are no other worlds, one could easily imagine a world where, say, the speed of light is different or where gravity works differently. We will denote by \definitie{possible worlds} these other worlds that either are or could have been. 
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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} model  is not precise enough. Let us restrict the \ghilimele{possible worlds} 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.}. 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.  It seems that reasoning about all the possible worlds could be very hard, but maybe we could do something easier, maybe we could reason about the mathematical axioms that model the worlds.  Let us consider the set of axioms that define a monoid. All groups are models for this set of axioms, but intuitively a group is something more interesting than a monoid and we should include extra axioms for defining it. On the other hand, if we relax the countability requirement, we could include all possible axioms for each model (e.g. for each monoid), uniquely identifying it, but again, intuitively this is not a useful way of modeling. We will call these the \ghilimele{too-general} \definitie{too-general}  problem and the \ghilimele{too-specific} \definitie{too-specific}  problem. For a given world, a \definitie{good set of axioms} would be one that would allow us to make all possible correct predictions for that world. The term prediction is not a clear one. To make it more clear, let us restrict again the \ghilimele{possible worlds} possible worlds  term. One option would be to make it to denote all possible worlds which have a concept of time and a concept of the state of the world at a given time and for which describing the state of the world at all possible times is equivalent to describing the world. 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 \ghilimele{the the  state of the world at a given time}, 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 \ghilimele{point} 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}. 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}. 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 \ghilimele{too-general problem} 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} 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 \ghilimele{infinite length set} infinite-length-set  notion is identical to \ghilimele{infinite set}.} 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 any binary encoding allowing us to represent these sets as binary strings, i.e. 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 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: 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 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. 
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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\footnote{All of these assume that the intelligent beings use a single axiom set for predicting. It could happen that they use multiple axiom sets which can't be merged into one. One could rewrite the paper to also handle this case, but it's easy to see that the finite/infinite distinction below would be the similar.}.  First, those intelligent beings could, at some point in time, find an axiom set 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 axiom set. We could relax this \ghilimele{best axiom set} condition by only requiring an also include here  axiom set sets  that is are  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. Second, those intelligent beings could reach a point where their theory clearly does not fully model the world, but it's also impossible to improve in a meaningful way. This could be the case if, e.g., they can model a part of their world, but modelling any part of the reminder would require adding an infinite set of axioms and no finite set of axioms would improve the model. 
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We should note that in the second and third cases it can also happen that one can’t improve the axiom set to cover more even when using a statistical axiom set. One such case would be when the perceived laws of the universe change in unpredictable ways from day to day (of course, this can happen without any change in the actual axiom set for the universe).  Let us denote by \definitie{observable description} of a possible universe $U$ for some intelligent beings $B$ inside $U$ any axiom set that, from the point of view of the beings $B$, can predict everything\footnote{With the assumptions above there is an axiom set that can predict everything. In general, an observable description would have to predict as much as it's possible.} that is reasonable\footnote{i.e. the \ghilimele{good enough} good enough  requirement from above.} about $U$ in the best way possible (i.e. predict precisely when possible or in a statistical way if not). 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}.} use 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 a chapter at the end with more mathematical justification for this. Maybe add there the 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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\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} 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 $t\ge 0$, 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}. 
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\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 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}. 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 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.