  This well founded path through the interpretability hierarchy has a canonical sequence through it which enables one to climb the hierarchy in a principled manner. At the base one has the very weak theory Q and one climbs upward by adding “closure principles”. This starts with principles asserting that certain fast-growing functions—exponentiation, super-exponentiation, etc.—are total and it proceeds upward through set existence principles, passing through the layers of second-order arithmetic, the subsystems of set theory, and the hierarchy of large cardinal axioms. 