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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 $\reale$. 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).  Note that a being living on Earth and ignoring everything outside of it may think that tides happen because gravity changes with time in ways which are more or less predictable, which seems similar to the case above. However, we also have another explanation available, i.e. that gravity works in the same way regardless of time, but the state of the universe, i.e. the relative position of the Sun, Moon and Earth, changes with time.  \section{Description probabilities}  For the purpose of this paper, let us denote by \definitie{finite property} of something any property of that something which can be written using a finite number of words. Since we will use only finite properties here, let us drop \ghilimele{finite} and call any of them simply \definitie{property}. 
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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. 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. [TODO: put this below and link it to the conclusion.]  [TODO: We experience gravity differently at various times and places - tides, variation from one place to another on Earth, on the Moon, when falling, although the law that describes gravitation does not change. We could imagine an universe where the actual law changes.]  [TODO: Do I need this paragraph? Where is the best place to put it? Probably I don't need it, but I should re-read the entire document: 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 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 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.