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Given a specific formalism for specifying axioms that uses a finite alphabet, for each possible world, we could only consider the smallest set of axioms that allow predictions, 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 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 that 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 \definition{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 \definition{optimal set of axioms}.  Now let us see if we actually need infinite length systems. We can have infinite systems of axioms, and there is no good reason to reject such systems and to ignore their possible worlds, so we will take them into account. It is less clear that we can't replace these infinite systems with finite ones. Indeed, let us use any binary encoding allowing us to represent these systems as binary strings, i.e. as binary functions over the set of natural numbers, i.e. $f:\naturale\longrightarrow \multime{0, 1}$. Then the following scenario becomes plausible: for any universe $U$ with an infinite system of axioms $A$, we can consider $U+encoding(A)$ to be an universe in itself. Then it's likely that we can find a finite system of axioms which describe $U+encoding(A)$. While, strictly speaking, this would be a different universe than the one we had at the beginning, it is also similar enough to it so one may be tempted to use only finite systems of axioms.  On the other hand, using only finite systems 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 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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Let us assume that if those intelligent beings could study their universe forever, they would try to improve their models in some essential way forever. Since they have an infinite time, they could use strategies like generating possible theories in 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 in this case the fraction of the world that can be modelled [TODO: Define separately] (if that notion makes sense) is increasing, but is limited, so it converges at some value. Also, the prediction error [TODO: define separately] 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 converge[S?] to different values then we are in the second case.  There is also a third case when there is no reasonable way to define the fraction of the world that can be modelled, except when the fraction is $0$. 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 planets would be close enough and would have enough resources (food, fuel) like food and fuel so  that humans would have no issues travelling between them. Light would have to come to them in a different way than in our world and something else, not gravitation would keep them in place. The laws of this hypothetical world, as observed by humans, would be both close enough to the laws in our world so that humans can live on any of the planets, but also different in an easily observable way. Let us say that, starting at 10 meters above ground, gravity would be described with a different function on each planet. On some planets it would follow an 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. [TODO: I had this comment: Will use infinite descr. instead of ... obser.]  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. If one would think that they cover a non-zero fraction because they cover a non-trivial part of the fundamental forces, then we could also vary all  theother  forces from one planet to the other or we could add other forces. a new set of forces for each planet.  The point is that one can't speak of we can have  a case when the  fraction of the world universe  that can be axiomatized at any moment  is modelled, zero,  even if one is able to model meaningful things (or, at least, about  the fraction universe and the model  is always 0). continuously extended.  [TODO: Make sure these things work for nondeterministic universes]  For the given intelligent beigns beings  we would say in the first case that their universe has a finite observable model description  and in the second and third case that it has an infinite observable model. description.  Of course, a possible universe $U$  could have multiple types of intelligent beings and beings, each type perceiving the universe in a different way. Because of this difference in perception, for  some would find types $U$ may have  a finite observable model and description while for  others would find it may have  an infinite observable model. description.  If we can have infinite descriptions, then the set of descriptions optimal systems of axioms  would have the same cardinality as the set of real numbers $\reale$. Indeed, for the planets-on-a-line example above,  we can find could select any subset of planets and get  an infinite universe with an optimal  set ofdisjoint sets of  axioms that are different enough and is distinct  fromwhich one can select  any infinite subset and join other subset. The set of all subsets of $\naturale$ has the same cardinality as $\reale$, so  thesubset to form another  set of optimal systems of  axioms which is predictive, has would have  at least a model and this cardinality. On the other hand, each system of axioms is written using an at most countable set of symbols, so there  can't be defined in a finite way [TODO: example or a better explanation. And a demonstration if needed]. more than $\reale$ systems of axioms.  These observable models 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 mathematical proposition property  $P$ we could try to see what is the chance that it's true ina random observable model. If we can agree on what "true" means, we could ask  the same thing about any natural language proposition. [TODO: use the terminology in "on the plurality set  of worlds"] observable descriptions.  In order to compute the probability of a proposition property  $P$ we would need a statistical distribution for the set of observable models. Unfortunately, we have no good way of chosing choosing  among the many possible distributions. Still, there is a class of distributions which stands out as being reasonable. 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 model description is zero. [TODO: use a consistent term for observable model/definition]