 The continuum problem is shown to be consistent with ZF by finding an enumeration of the reals which is indexed by the countable ordinals, a strategy which had been recognized as a promising one already by  Hilbert.[15]  The problem, and the intuition behind the proof, is to build a “small” model, one in which the absolute minimum number of reals is allowed, while at the same time the model is large enough to be closed under all the operations the ZF axioms assert to exist.