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Now let us see if we actually need infinite length sets. We can have infinite sets\footnote{Note that for a given finite alphabet, i.e. in the current context, the infinite-length-set notion is identical to 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 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 sets of axioms.  On the other hand, using only finite sets of axioms in this way seems to be some sort of cheating. In order to get a more honest system of axioms, we should define the state of the universe as what could be changed from inside. As an alternative, we  could work with very specific 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 set encoding problem by talking However, not all universes are interesting, so let us focus  only about on  worlds which could contain intelligent beings and talking let us think  about how the intelligent beings would model their world. Even more, let us include only world where there intelligent beings would be similar enough to us in that they can use logic and mathematics, but they wouldn't be able to process an infinite amount of that in a finite time. Let us note that in worlds with an infinite optimal set of axioms these intelligent beings they will never have a complete description of how their world works, but they will be able to build better and better models. Let us assume that those intelligent beings are continuously trying to find better models for their world and that they are reasonably efficient at this. 
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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 - I explained it.]  [TODO: Use can't, won't, isn't isn't, let's  and can not, will not, is not not, let us  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?]  [TODO: define "mathematical prediction" or something like that and use it here, since I have two meanings of prediction - I think this is done/not needed, I added a footnote in the only place where I use the normal meaning of prediction. I have to check. Maybe I should be more explicit about this.]