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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.]  \section{Description probabilities}  These observable descriptions of possible worlds are general enough and different enough that it's hard to say something about them, except that they make sense in a mathematical way. Still, given any property $P$ we could try to see what is the chance that it's true in the set of observable descriptions.  If our universe is not created [TODO: designed??], then any possible universe could have existed (and maybe all possible universes actually exist). Focusing only on universes which have a space-time and in which intelligent beings can exist, if we would want to pick a random one for a reasonable definition of random, each universe would have a zero probability of being chosen. If we further 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 to be randomly picked as any other, so each has a zero probability. I argue that, even more, the systems of axioms that would be produced by the intelligent beings in that universe (in the sense mentioned above) have each a zero probability. In other words, any reasonable probability over these axiom systems is continuous.