n arrow " ▁hierarchy ▁which ▁gives ▁the ▁construct ible ▁universe ▁L , ▁which ▁also ▁satisfies ▁all ▁the ▁ax i oms ▁of ▁Z FC , ▁including ▁the ▁ax iom ▁of ▁choice . ▁It ▁is ▁independent ▁from ▁the ▁Z FC ▁ax i oms ▁whether ▁V   =   L . ▁Although ▁the ▁structure ▁of ▁L ▁is ▁more ▁regular ▁and ▁well ▁behav ed ▁than ▁that ▁of   V , ▁few ▁mathemat icians ▁argue ▁that   V ▁=   L ▁should ▁be ▁added ▁to ▁Z FC ▁as ▁an ▁additional ▁" ax iom ▁of ▁construct ibility ". ▁ ▁Met am ath emat ics ▁ ▁Virtual ▁classes ▁ ▁As ▁noted ▁earlier , ▁proper ▁classes ▁( collections ▁of ▁mathematical ▁objects ▁defined ▁by ▁a ▁property ▁shared ▁by ▁their ▁members ▁which ▁are ▁too ▁big ▁to ▁be ▁sets ) ▁can ▁only ▁be ▁treated ▁indirect ly ▁in ▁Z F ▁( and ▁thus ▁Z FC ). ▁An ▁alternative ▁to ▁proper ▁classes ▁while ▁stay ing ▁within ▁Z F ▁and ▁Z FC ▁is ▁the ▁virtual ▁class ▁not ational ▁construct ▁introduced ▁by ▁, ▁where ▁the ▁entire ▁construct ▁y ▁ ∈ ▁{ ▁x ▁| ▁F x ▁} ▁is ▁simply ▁defined ▁as ▁F y . ▁ ▁This ▁provides ▁a ▁simple ▁notation ▁for ▁classes ▁that ▁can ▁contain ▁sets ▁but ▁need ▁not ▁themselves ▁be ▁sets , ▁while ▁not ▁comm itting ▁to ▁the ▁ont ology ▁of ▁classes ▁( because ▁the ▁notation ▁can ▁be ▁s ynt act ically ▁converted ▁to ▁one ▁that ▁only ▁uses ▁sets ). ▁ ▁Qu ine ' s ▁approach ▁built ▁on ▁the ▁earlier ▁approach ▁of ▁. ▁ ▁Virtual ▁classes ▁are ▁also