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Let us consider the set of axioms that define a monoid. All groups are models for this set of axioms, but intuitively a group is something more interesting than a monoid and we should include extra axioms for defining it. On the other hand, we could include all possible axioms for each model (e.g. for each monoid), uniquely identifying it, but again, intuitively this is not a useful way of modeling. We will call these the "too-general" problem and the "too-specific" problem.  For a given world, a good set of axioms would be one that would allow us to make all possible correct  predictions for that world. The term prediction is not a clear one. To make it more clear, let us restrict again the "possible worlds" term. One option would be to make it to denote all possible worlds that/which? have a concept of time and a concept of the state of the world at a given time and for which describing the state of the world at all possible times is equivalent to describing the world. This ignores some important issues like the fact that it's reasonable to have a concept of time without having a well defined concept of "the state of the world at a given time", so we could rephrase the definition above to include many other reasonable notions of space and time, e.g. we can include worlds where "point" is a concept and we can know which pairs (point, time) are before a given (point, time) pair.  Then when we can say that we can \definitie{predict} something means that we have a system of axioms for which the state of the world (maybe at a given point) is uniquely determined by the state of the world at a subset of the previous points in time\footnote{Will be extended to statistical predictions in the next paragraph}. If we are interested in predicting the state at a given point $P$, this subset should include a full section through $P$'s past (e.g. a plane which intersects it's past cone), i.e. it should separate $P$'s past in two parts, one which is "before the subset" and one which is "after the subset"\footnote{This means that all lines which fully lie in $P$'s past and connect a point which is before the subset with a point which is after the subset must go through the subset}. One could think of similar definitions for predicting the entire state of the world. If needed, this definition could be changed to work for more concepts of space and time.  In a deterministic universe, knowing the laws of the universe and its full state we could, in theory, fully predict $P$'s state. But an universe does not have to be deterministic and, even if it is, one could have only a statistical model for it. Then we will allow using  a set of axioms which only allows gives a  statistical predictions distribution for the state of the universe given its past  (I'll call this a \definitie{statistical axiom set}) is fine and for set}). For  the purpose of this document we don't need to make a difference between a non-deterministic universe and a deterministic one but for which we only have a statistical model.If we define "prediction" in some useful way, as suggested above, and restrict the "possible worlds" term to the ones where we can make predictions, then it makes sense to use only systems of axioms that allow predictions. We will always use the best possible system for predictions (statistical or not). This solves the "too-general problem" since such a system would describe its possible worlds in a way that does not leave out important things. Still, there is nothing that prevents such a system from going into too much details.  Let us restrict the "possible worlds" term to the worlds where we can make predictions and let us use only systems of axioms that allow predictions. As mentioned above, for a given world, a good set of axioms is one which allows us to make all possible correct predictions for that world (statistical or not). Using only good sets of axioms solves the "too-general problem" since such a system would describe its possible worlds in a way that does not leave out important things. Still, there is nothing that prevents such a system from going into too much details.  Given a specific formalism for specifying axioms that uses a finite alphabet, for each possible world, we could only consider say that  the smallest best [TODO: Not working with maximal]  set of axioms that allow predictions, is the smallest good one,  smallest being defined as "having the smallest length when written on paper". This is not a well defined notion for a few reasons. First, there could be multiple systems with "the smallest length" (one obvious case is given by reordering the axioms). In such a case, we could define an order for the symbols that we are using in our formalism and we could pick the system with the smallest length and that which  is the smallest in the lexicographic order. Second, there could be systems of axioms of infinite length. For this, we will only consider systems which, when written on an infinite paper, use a countable number of symbols. This means that all will have the same length, but we can still use the lexicographic order to compare them. We will ignore systems which need an uncountable set of symbol places. With an axiom system chosen in this way we would also solve the "too-specific problem" since we would remove any axiom that's not absolutely needed. 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. We can have infinite systems of axioms, and there is no good reason to reject such systems 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 with finite ones. Indeed, let us use any binary encoding allowing us to represent these systems as binary strings, i.e. as binary functions over the set of natural numbers, i.e. the encoding of a set of axioms $A$, $encoding(A)$, would be a function  $f:\naturale\longrightarrow \multime{0, 1}$. Then the following scenario becomes plausible: for any universe $U$ with an infinite system of axioms $A$, we can consider $U+encoding(A)$ to be an universe in itself. 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 describe $U+encoding(A)$. allows predictions for such an universe.  While, strictly speaking, this would be a different universe than the one we had at the beginning, it is also 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.  \section {Intelligent beings / Finite and infinite axiom systems [but then I would need to move the title above.]}  We could also completely avoid this the axiom system encoding  problem by talking only about worlds which could contain intelligent beings and talking about how the intelligent beings would model their world. This approach may also be a more interesting one, since we may never have a complete and correct model for our world, but we can build better and better models. As a parenthesis, note that until now we restricted the "possible world" concept several times. The argument below also works with larger "possible world" concepts as long as those worlds have a few basic properties (e.g. one can make predictions and it can contain intelligent beings) and at the same time it is plausible that our world is such a "possible world".