We describe the order on the ratios that define the generic universal smooth period doubling Cantor set. We prove that this set of ratios forms itself a Cantor set, a Conjecture formulated by Coullet and Tresser in 1977. We also show that the two period doubling renormalization operators, acting on the codimension one space of period doubling maps, form an iterated function system whose limit set contains a Cantor set.