 We noted above that the second incompleteness theorem provides us with a case where the statement, namely, Con(T), leads to a jump in the interpretability hierarchy while its negation does not. There are also natural examples of this phenomenon—for example, large cardinal axioms. When a statement ϕ is independent of a theory T in this manner—that is, in virtue of the fact that T &lt; T + ϕ—we shall refer to the kind of independence involved as vertical independence. Thus, the second incompleteness theorem demonstrated that Con(PA) is vertically independent of PA (assuming, of course, that PA is consistent) and that the large cardinal axioms “There is a strongly inaccessible cardinal” is vertically independent of ZFC (assuming that ZFC + “There is a strongly inaccessible cardinal” is consistent). 