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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).  Let us denote by \definitie{observable description} of a possible universe $U$ for some intelligent beings $B$ inside $U$ any axiom set that can make all possible predictions for $U$ from the point of view of the beings $B$.  [TODO: Make sure these things work This is not a good definition, I should clarify what happens if there can be a universe $U$ in which there are predictions that can't be covered by any countable axiom set. Maybe I should use only universes  for nondeterministic universes] which I can make those, but then ours may not be such a universe. Or maybe I should use the paragraph below as a definition.]  For Then, for  the given intelligent beings we would say in the first case that their universe has a finite \definitie{finite  observable description description}  and in the second and third case that it has an infinite \definitie{infinite  observable description. 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 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.] here. Maybe I shouldn't bother.]  \section{Description probabilities}  These observable descriptions For the purpose of this paper, let us denote by \definitie{finite property}  ofpossible worlds are general enough and different enough that it's hard to say  somethingabout them, except that they make sense in a mathematical way. Still, given  any property $P$ we could try to see what is the chance of  that it's true in the set something which can be written using a finite number of words. Since we will use only finite properties here, let us drop "finite" and call any  of observable descriptions. them simply \definitie{property}.  If our universe is not designed, then any possible universe could have existed (and maybe all These observable descriptions of  possible universes actually exist). Focusing only on universes which have a space-time worlds are general enough  and in which intelligent beings can exist, if we would want different enough that it's hard  to pick a random one for a reasonable definition of random, each universe would have say something about them, except that they make sense in  a zero probability of being chosen. If mathematical way. Still, given any property $X$  wefurther 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 try  to be randomly picked as any other, so each has a zero probability. I argue that, even more, see what is  the systems of axioms chance  that would be produced by the intelligent beings it's true  inthat universe (in  the sense mentioned above) have each a zero probability. In other words, any reasonable probability over these axiom systems is continuous. set of observable descriptions.  [TODO: Delete if I am happy with the paragraph before this one. In order to compute the probability of 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 property $P$ space-time and in which intelligent beings can exist, if  we would need want to pick  a statistical distribution random one  for the set a reasonable definition  of observable descriptions. We random, each universe would  have no good way a zero probability of being chosen. If we further restrict these universes to ones which allow a predictive system  of choosing among axioms for  the many possible distributions, entire universe\footnote{The entire universe is required here for simplicity,  but we one could also work when only a part of the universe  can still find interesting things without much choosing. Let us note that there is have  a class predictive system  of distributions which stands out as being reasonable for this purpose: axioms.}, then each system of axioms is  as long likely to be randomly picked  aswe don't see  any reason to prefere other, so each has  a specific description over all zero probability. I argue that, even more,  the others, our only choice is to use continuous distributions, systems of axioms  that is, distributions for which would be produced by  the probability of intelligent beings in that universe (in the sense mentioned above) have each a zero probability. In other words,  any given observable description reasonable probability distribution over these axiom systems  is zero.] continuous.  [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 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.  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 Now, it could happen  that for any description there is  a zero-probability  property that which  is true for only one description has a zero probability (i.e. that description, making  it likely that we can't find anything interesting this way. Fortunately, this  is false virtually everywhere). Even more, any not true. Indeed, a  propertywhich  is true for written using  a countable number of descriptions finite alphabet and  has a zero probability. This means that any property with a non-zero probability finite length, so there  is for sure true for an uncountable at most a countable  number of descriptions. Of course, there may such properties. Let $Y$  be properties which are true for an uncountable number this set. Then $P(Y)$, the probability of the set  of observable  descriptions and still have a zero probability. for which at least one proposition in $Y$ is true, is the sum of the probabilities of all elements in $Y$, so $P(Y) = 0$.  Now, it could happen We can then say  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 virtually all descriptions [TODO: Make sure I want descriptions here and  not optimal axiom sets or something. Probably I want descriptions], only properties with non-zero probability are  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 $P(Y)$, This means that, if  the probability of the set of observable models for which at least one proposition in $Y$ our world being designed  is true, is the sum of non-zero,  the probabilities of all elements in $Y$, so $P(Y) = 0$. only rational choices are that either our world is designed or only non-zero probability properties are true.  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 designed is non-zero, the only rational choices are that either our world is designed or only non-zero probability properties are true. Now, let us return to the issue of observable models descriptions  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 property [TODO: is it a property of the description or of the universe? Does that match what I said above about only inferring things about descriptions? Ah, it's a property of an axiom set, I think.]  is a zero-probability one, so either our universe is designed, or at any point in time there will be aninfinite 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 model no matter how hard  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?] try.  \section{Approximations}