I don't believe it's good math until it becomes possible to talk about the first uncountable ordinal, in the way that you can talk about the integers. Any first-order theory of the integers, like first-order PA, will have some models containing supernatural numbers, but there are many different sorts of models of supernatural numbers, you couldn't talk about the supernaturals the way you can talk about 3 or the natural numbers. My skepticism about "the first uncountable ordinal" is that there would not exist any canonicalizable mathematical object - nothing you could ever pin down uniquely - that would ever contain the first uncountable ordinal inside it, because of the indefinitely extensible character of well-ordering. This is a sort of skepticism of Platonic existence - when that which you thought you wanted to talk about can never be pinned down even in second-order logic, nor in any other language which does not permit of paradox.