  A standard model of Z2 has the form Vκ where κ is a regular cardinal such that for all α &lt; κ, 2α &lt; κ.  Such a cardinal is called a (strongly) inaccessible cardinal.  The next axiom in the hierarchy under consideration is the statement asserting that such a cardinal exists.  The resulting theory   ZFC + “There is a strongly inaccessible cardinal” proves that there is a level of the universe that satisfies ZFC.  Continuing in this fashion one arrives at stronger and stronger axioms that assert the existence of larger and larger levels of the universe of sets.  Before continuing with an outline of such axioms let us first draw the connection with the hierarchy of interpretability.