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 make do justice to 2 in a few lines). I have doubts if I should make this one community wiki.

1

# Martin's "Philosophical Issues about the Hierarchy of Sets"

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 make justice to 2 in a few lines). I have doubts if I should make this one community wiki.