  One can pass beyond the above hierarchy of sets of reals by considering richer notions of definability. We shall say little about these stages, except to note that there are natural limits to the lightface and boldface hierarchy, namely that of being definable from ordinal parameters (OD) and that of being definable from ordinal parameters and real numbers (OD(ℝ)). To see that the notion of ordinal definability is a limiting case of the lightface hierarchy notice first that any notion of definability which does not render all of the ordinals definable can be transcended (as can be seen by considering the least ordinal which is not definable according to the notion) and second that the notion of ordinal definability cannot be so transcended (since by reflection OD is ordinal definable). (Similar considerations apply to the boldface hierarchy.) It is for this reason that Gödel proposed the notion of ordinal definability as a candidate for an “absolute” notion of definability. As a limiting case of the lightface and boldface hierarchy we therefore have the sets of reals that are OD and the sets of reals that are OD(ℝ), respectively. 