deletions | additions       diff --git a/untitled.tex b/untitled.tex  index 01ae6a0..b513763 100644  --- a/untitled.tex  +++ b/untitled.tex 
...
\item As the fraction of the optimal set of axioms that is implied by the current axiom set.  \item As something between the first two cases, where we use a weighted fraction of the optimal axiom set, each axiom having a weight proportional to the fraction of the world where it applies. As an example, let us say that we have an infinite set of axioms, and that for each point in space we can predict everything that happens using only three axioms (of course, two different points may need different axiom triplets). Let us also assume that there is a finite set of axioms $S$ such as each point in space has an axiom in $S$ among its three axioms. Then $S$ would model at least a third of the entire space.  \end{enumerate}  In all of these cases, the predictions made only from artificial constraints imposed by this paper, e.g. that the world can be modelled mathematically or that it contains intelligent beings, should not count towards the fraction of the world that is modelled by an axiom set. In other words, this \definitie{fraction of the world} is actually the fraction of the world that is modelled without above  what is absolutely needed because of the constraints imposed here. We can use any of these definitions (and many other reasonable ones) for the reminder of this paper. Then we would have three possible cases\footnote{All of these assume that the intelligent beings use a single axiom set for predicting. It could happen that they use multiple axiom sets which can't be merged into one. One could rewrite the paper to also handle this case, but it's easy to see that the finite/infinite distinction below would be the similar.}.