 The above limitative results show that ZFC lacks sufficient power to resolve many questions about simple definable sets of reals—it cannot determine whether Σ̰12 sets have the regularity properties and it cannot determine the pattern of the uniformization property beyond Σ̰12. This placed mathematicians in an interesting predicament. In order to resolve these questions they must expand their resources beyond ZFC. But there are many ways of doing this and it is unclear at the outset whether there is a correct direction of expansion. 