Some months ago (October 2010), in the context of the Workshop on Set Theory and the Philosophy of Mathematics, Professor Donald A. Martin gave a talk entitled "Philosophical issues about the hierarchy of sets".
Abstract: I will discuss some philosophical questions about the cumulative hierarchy of sets, its levels, and their theories. Some examples:
(1) It is sometimes asserted one cannot quantify over everything. A related assertion is that each of our statements about the universe of sets can from a different perspective be seen as a statement about some Va. Thus the class-set distinction is really a relative one. Does this make sense? Is it right?
(2) Is the first order theory of V determinate? Does every sentence have a truth value? Are there levels of the hierarchy whose first order theories are indeterminate? If so, what is the lowest such level? What about L and the constructibility hierarchy?
(3) There are many examples of proofs of a statement about one level of the hierachy that use principles about a higher level. Under what conditions and in what sense do these count as establishing the lower level statement?
I will discuss these questions mainly from a viewpoint that takes mathematics to be about basic mathematical concepts, e.g., those of natural number, real number, and set.
I am highly interested in learning how these questions might be answered (as you may problably know from previous questions of mine here in MO), so I would be grateful if anyone could give any information in this respect, especially for those questions of 1 and 3 (I am afraid it is almost impossible to do justice to 2 in a few lines).