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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, we will only consider 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. [TODO: Can I pick a minimum using inclusion? Can I use the fact that a finite set of axioms implies a larger set and replace them?]. With an axiom set chosen in this way we would also solve the \ghilimele{too-specific problem} in the finite case. In the infinite case we 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.  Note that for a given finite alphabet the \ghilimele{infinite length set} notion is identical to \ghilimele{infinite set}.  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}.  Now let us see if we actually need infinite length sets. We can have infinite sets sets\footnote{Note that for a given finite alphabet, i.e. in the current context, the \ghilimele{infinite length set} notion is identical to \ghilimele{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 system 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 systems sets  of axioms. On the other hand, using only finite systems 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 systems 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. \section {Modelling from inside}  We could also completely avoid the axiom system set  encoding problem by talking only about worlds which could contain intelligent beings and talking about how the intelligent beings would model their world. Let us note that if those intelligent beings are similar enough to us and the optimal set of axioms for their world is infinite then they will never have a complete description of how their world works, but they will be able to build better and better models. We will assume that those intelligent beings are continuously trying to find better models for their world and that they are reasonably efficient at this.  As a parenthesis, note that until now we restricted the possible world concept several times. The argument below also works with larger possible world concepts as long as those worlds have a few basic properties (e.g. one can make predictions and it can contain intelligent beings) and at the same time it is plausible that our world is such a possible world.  First, let us note that having intelligent beings in an universe likely means that their intelligence is needed to allow them to live in that universe, which likely means that they can have a partial model of the universe. That model does not have to be precise, e.g. it could be made of simple rules like \ghilimele{If I pick fruits then I can eat them. If I eat them then I live}, and it can cover only a small part of their world, but it should predict\footnote{This is the only place where predict means that the beings can actually say something about the future instead of a theoretical way of making predictions. Everywhere else we're using the previous definition of prediction which only requires that prediction is possible for a being which can take full snapshots of the universe and can go through all the possible models of an axiom system.} set.}  something. Of course, these predictions do not have to be deterministic. Also, they might not be able to perceive the entire universe. 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 set  in at least three ways: \begin{enumerate}  \item As the fraction of the observable space for which the axiom system set  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. set.  \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}  In all of these cases, the 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 \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\footnote{All of these assume that the intelligent beings use a single axiom system set  for predicting. It could happen that they use multiple axiom systems 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 system 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 system. axiom set.  We could relax this \ghilimele{best axiom system} set}  condition by only requiring an axiom system set  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. 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 get one a better model.  In order to make this split into cases more clear, let us assume that those intelligent beings would study their universe and would try to improve their axiom systems sets  in some essential way forever. Since they have infinite time available to them, they could use strategies like generating possible theories in order (using the previously defined order), checking if they seem to make sense and testing their predictions against their world, so let us assume that if there is a possible improvement to their current theory, they will find it at some point. Note that the fraction of the world that can be modelled is increasing, but is limited, so it converges at some value. Also, the prediction error (it's not important to define it precisely here) is decreasing and is limited, so it converges. If the fraction converges at $1$ and the prediction error converges at $0$, then we are in the first case, because we reach a point when the fraction is so close to $1$ and the error is so close to $0$ that one would find them good enough. If the fraction or the error converges to different values then we are in the second case.  There is also a third case, when one can improve the axiom system set  in ways that seem meaningful, without growing the fraction of the world that is covered by the system set  and without decreasing the prediction error. As an example, imagine a world with an infinite number of earth-like planets that lie on one line and with humans living on the first one. The laws of this hypothetical world, as observed by humans, would be wildly different from one planet to the other. As an example of milder differences, starting at $10$ meters above ground, gravity would be described with a different function on each planet. On some planets it would follow the inverse of a planet-specific polynomial function of the distance, on others it would follow the inverse of an exponential function, on others it would behave in some way if the distance to the center of the planet in meters is even and in another way if the distance is odd, and so on. Let us also assume that humans can travel between these planets freely in some bubble that preserves the laws of the first planet well enough that humans can live, but that also lets them observe what happens outside. In this case one could study each planet and add a specific description of the laws for each, but at any moment in time the humans in this world would only have a finite part of an infinite set of laws, so we wouldn't be able to say that they cover a non-zero fraction of the laws or a non-zero fraction of the world. If one would think that they cover a non-zero fraction because (say) they cover a non-trivial part of the fundamental forces, then we could also vary the type of all forces from one planet to the other or we could add a new set of forces for each planet. The point is that we can have a case when the fraction of the universe that can be axiomatized at any moment is zero and one can't improve this fraction, even if one is able to model new meaningful things about the universe and the part of the world that is covered by the axiom system set  is continuously extended. We should note that in the second and third cases it can also happen that one can’t improve their 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). 
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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 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 $\reale$. 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 systems sets  of axioms would have at least this cardinality. On the other hand, each system set  of axioms is written using an at most countable set number  of symbols, symbols over a finite alphabet,  so there can't be more than $\reale$ systems 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.] \section{Description probabilities} 
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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 $X$ we could try to see what is the chance that it's true in the set of observable descriptions.  If our universe is not 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 set  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.}, universe,  then each system set  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 sets  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 distribution over these axiom systems sets  is continuous. Above I required that there is a predictive axiom set for the entire universe. That was done for simplicity, but a similar parallel construction could be made for the case when only a part of the universe can have a predictive set of axioms.  While this is designed such that we can't directly say anything about a specific observable description, we can say things about what has a real chance of being true for a random observable description. First, let us note that any property that is true for only one description has a zero probability (i.e. it is false virtually everywhere). Even more, any property which is true for a countable number of descriptions has a zero probability. This means that any property with a non-zero probability is for sure true for an uncountable number of descriptions. Of course, there may be properties which are true for an uncountable number of descriptions and still have a zero probability. 
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[TODO: Make sure I'm using quotes correctly and consistently.]  [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: 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?]