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Let us chose any formalism for specifying axioms that uses a finite alphabet and let us choose an arbitrary order relation on that alphabet. Also, we will only consider axiom sets which, when written on an infinite paper, use at most a countable number of symbols. Then, for each possible world, we could say that the best set of axioms is the smallest good one, smallest being defined as having the smallest length when written on (possibly infinite) paper. This is not a well defined notion, so let us also say that for sets of axioms having finite length we will break any ties by using the lexicographic order. For worlds having only sets of axioms of infinite length we can't use the same trick to break ties, but they all have the same length (countably infinite) so we will choose any such set of axioms and say that it's the best one. [TODO: Can I pick a minimum using inclusion? Can I use the fact that a finite set of axioms implies a larger set and replace them?]. With an axiom set chosen in this way we would also solve the \ghilimele{too-specific problem} in the finite case. In the infinite case we made an arbitrary decision, but then the cardinality of the axiom set matches the complexity of the world anyway, even if there is some redundancy.  Note that for a given finite alphabet the \ghilimele{infinite length set} notion is identical to \ghilimele{infinite set}.  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 \definitie{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 \definitie{optimal set of axioms}.  Now let us see if we actually need infinite length systems. sets.  We can have infinite systems sets  of axioms, and there is no good reason to reject such systems 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 systems sets  with finite ones. Indeed, let us use any binary encoding allowing us to represent these systems sets  as binary strings, i.e. as binary functions over the set of natural numbers. 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 system 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 system 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 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.