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An This is an  attempt to reason about why our world is the way it is. is and what we can reasonably believe about it.  Many people believe that the world is designed and  created. However, this belief is 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 the creator of this world as "the Creator" is reasonable.} wanted it to have certain properties so in 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 this path it  here. Let us consider the other case. If the our  world was not designed and  created, 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 aren't are no  other worlds, ours could have been different. We will denote by \definitie{possible worlds} these other worlds that either are or could have been. \section{Modelling possible worlds}  First, let us note that there can't be any causal interaction between  two different possible worlds should not interact between them. worlds.  If they two worlds  are interacting, it's more reasonable to say that they are actually a single possible world with two parts. How would a possible world look like? It could have exactly the same fundamental laws as ours, but with the matter organized differently. It could have similar laws, but with different universal constants. It could have different fundamental particles (or whatever the basic building blocks of our universe are, assuming that there are any). Or it could be completely different, i.e. different in all possible ways.  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 "model" is not precise enough. Let us restrict the "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 non-contradictory, 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.} set of axioms models some 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 there isn't any designer for 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 "too-general" problem and the "too-specific" problem. For a given world, a 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 "possible worlds" term. One option would be to make it to denote all possible worlds that/which? 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 "the state of the world at a given 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 "point" is a concept and we can know whichpairs  (point, time) pairs  are before a given (point, time) pair. Then when Then, in the following,  we can will  say that we can \definitie{predict} something means that ($S$) whenever  we have a system set  of axioms for which the state of the world (maybe at a given point) $S$  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 $P$, $P$ and time $t$, a good choice for  this subset should include could be  a full section through $P$'s past (e.g. a plane which intersects it's past cone), i.e. it should separate 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, knowing if we know  the laws of the universe and its full state at a given time,  we could, in theory, fully predict $P$'s any future  state. But an universe does not have to be deterministic and, even if it is, one could have only a statistical model for it. 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 systems sets  of axioms that allow predictions. As mentioned 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 "too-general problem" since such a system set  would describe its possible worlds in a way that does not leave out important things. Still, there is nothing that prevents such a system set  from going into too much details. Given a specific formalism for specifying axioms that uses a finite alphabet, for each possible world, we could say that the best [TODO: Not working with maximal] set of axioms is the smallest good one, smallest being defined as "having the smallest length when written on paper". This is not a well defined notion for a few reasons. First, there could be multiple systems sets  with "the smallest length" (one obvious case is given by reordering the axioms). In such a case, we could define an order for the symbols that we are using in our formalism and we could pick the system with the smallest length and which is the smallest in the lexicographic order. Second, there could be systems of axioms of infinite length. For this, we will only consider systems which, when written on an infinite paper, use a countable number of symbols. This means that all will have the same length, but we can still use the lexicographic order to compare them. We will ignore systems which need an uncountable set of symbol places. With an axiom system chosen in this way we would also solve the "too-specific problem" since we would remove any axiom that's not absolutely needed. If $U$ is an universe and $A$ is the smallest set of predictive axioms as described 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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These observable descriptions of possible worlds are general enough and different enough that it's hard to say something about them, except that they make sense in a mathematical way. Still, given any property $P$ we could try to see what is the chance that it's true in the set of observable descriptions.  If our universe is not created [TODO: designed??], designed,  then any possible universe could have existed (and maybe all possible universes actually exist). Focusing only on universes which have a space-time and in which intelligent beings can exist, if we would want to pick a random one for a reasonable definition of random, each universe would have a zero probability of being chosen. If we further restrict these universes to ones which allow a predictive system of axioms for the entire universe\footnote{The entire universe is required here for simplicity, but one could also work when only a part of the universe can have a predictive system of axioms.}, then each system of axioms is as likely to be randomly picked as any other, so each has a zero probability. I argue that, even more, the systems of axioms that would be produced by the intelligent beings in that universe (in the sense mentioned above) have each a zero probability. In other words, any reasonable probability over these axiom systems is continuous. [TODO: Delete if I am happy with the paragraph before this one. In order to compute the probability of a property $P$ we would need a statistical distribution for the set of observable descriptions. We have no good way of choosing among the many possible distributions, but we can still find interesting things without much choosing. Let us note that there is a class of distributions which stands out as being reasonable for this purpose: as long as we don't see any reason to prefere a specific description over all the others, our only choice is to use continuous distributions, that is, distributions for which the probability of any given observable description is zero.] 
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Now, it could happen that for any description there is a zero-probability property which is true for that description, making it likely that we can't find anything interesting this way. Fortunately, this is not true. Indeed, a property is written using a finite alphabet and has a finite length, so there is at most a countable number of such properties. Let $Y$ be this set. Then $P(Y)$, the probability of the set of observable models for which at least one proposition in $Y$ is true, is the sum of the probabilities of all elements in $Y$, so $P(Y) = 0$.  We can then say that for virtually all models, only properties with non-zero probability are true. This means that, if the probability of our world being created designed  is non-zero, the only rational choices are that either our world is created designed  or only non-zero probability properties are true. Now, let us return to the issue of observable models being finite or infinite. With an finite alphabet, only a countable set of models have a finite observable description. Then the "has a finite description" proposition is a zero-probability one, so either our universe is created, designed,  or at any point in time there will be an infinite number of things that we didn't axiomatize yet about our universe but we think that they are important. [TODO: It's probably better to say that there will be an important part of our universe that we can observe but can't model. Also, since we chosed the precision and prediction error in an arbitrary way, this part that can't be modelled is visible at any "zoom" level.] [TODO: Find the right term for "has a finite description" thing. Is it property? Is it proposition? How are these terms used in philosophy?] 
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\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 $\epsilon \ge 0$ and a probability $q\ge 0$ for our predictions [TODO: did I define this?] and let us denote by $f$ with $0 < f \le 1$ the fraction of the world where we can make predictions using the given time length $t$, the acceptable error $\epsilon$, having a probability $p$ that the prediction is correct.  Then, if the world is not created, 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), $\epsilon = \infty$ (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: Do I need this paragraph? Where is the best place to put it? 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 of the observers (assuming that the universe has a concept of place that is close enough to ours). The laws of the universe as observed at a given time and place can be quite different from the laws that one can observe 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). Note that these beings must be able to live through the changes, otherwise the universe does not count for our problem.]  From the above, we have two options. Either our universe is created designed  and then we might be able to make predictions for a non-trivial part of the universe we can observe (assuming that we have enough details about the state of the universe), or the universe is not created designed  and then, although we can make predictions for a small part of the universe, we can't make predictions outside of it, no matter how much information about the state of the universe we would have; also, this small part would be an insignificant fraction of what we could observe. We seem to be able to make predictions for mostly everything that we can observe, even if we may not be able to make many predictions for very distant things. We also have no sign that the laws of the universe would be significantly different outside of Earth, so it seems that the limiting factor is that we don't know the state of the universe. Then the second option is probably false and the first one is probably true.