for any finite ordinal n, the sets l and v are the same (whether v equals l or not), and thus l = v: their elements are exactly the hereditarily finite sets. equality beyond this point does not hold. even in models of zfc in which v equals l, l is a proper subset of v, and thereafter l is a proper subset of the power set of l for all α > ω. on the other hand, v equals l does imply that v equals l if α = ω, for example if α is inaccessible. more generally, v equals l implies h equals l for all infinite cardinals α.