  We have already seen one way of climbing the hierarchy of the degrees of interpretability, namely, by adding consistency statements.  There are two drawbacks to this approach.  First, if one starts with a theory that “arises in nature” and adds the consistency statement one lands in a degree that has no known representative that “arises in nature”.  Second, the consistency statement does not take one very far up the hierarchy.  Both of these drawbacks are remedied by a very natural class of axioms—the large cardinal axioms.