 Let us stress from the beginning that because of the incompleteness theorems the case for any new axiom is going to have to be delicate. The standard way of establishing a proposition in mathematics is by proving it within some accepted set of axioms. However, the axioms implicitly accepted by mathematicians in everyday practice seem to be exhausted by those of ZFC, and we are here dealing with questions and new axioms that are beyond the reach of ZFC. Given the nature of the case, it seems unlikely therefore that one will have a knock-down argument for new axioms. Instead it seems that the most that one can hope for is to accumulate evidence until eventually the case becomes compelling. The other alternative is that the case remains divided between two equally compelling but mutually inconsistent approaches. 