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In order to make this split into cases more clear, let us assume that those intelligent beings would study their universe forever and, if needed, they would try to improve their axiom systems in some essential way forever []. Since they have an infinite time, 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 in this case the fraction of the world that can be modelled is increasing, but is limited, so it converges at some value. Also, the prediction error [TODO: define separately – after I finish rewriting the definition should be towards the end of the document and should be moved before this paragraph.] 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.  [TODO: I stopped rewriting here.] There is also a third case case,  when there is no reasonable way to define one can improve the axiom system in ways that seem meaningful, without growing  the fraction of the world thatcan be modelled, except when the fraction  is $0$. covered by the system 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 planets would be close enough and would have enough resources 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, As an example,  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. [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. laws or a non-zero fraction of the world.  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 varyall  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, zero and one can't improve this fraction,  even if one is able to model new  meaningful things about the universe and the model part of the world that is covered by the axiom system  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 fully unpredictable ways from day to day (of course, this can happen without any change in the actual axiom set for the universe).  [TODO: Make sure these things work for nondeterministic universes]  For the given intelligent beings we would say in the first case that their universe has a finite observable description and in the second and third case that it has an 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 types 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 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 of axioms would have at least 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 more than $\reale$ systems 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.]  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??], 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. zero.]  [TODO: I stopped rewriting here.]  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 in a random observable description. First, let us note that a 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.