Modal Operators

Notwithstanding her rejection of quantification over an absolutely comprehensive domain, a relativist about quantifiers may still be tempted to seek other means to generalize. This chapter concerns relativist-friendly modal operators. By modalizing her quantifiers, the relativist has a systematic way to attain absolute generality, which permits her to regiment her view with a single modal formula, and to frame an attractive modal axiomatization of the iterative conception of set. In addition to the immediate cost of admitting the relevant modality into her ideology, however, this approach leads to a hybrid version of relativism, which has some significant commonalities with absolutism about quantifiers.

THE ITERATIVE CONCEPTION OF SET

The two expressions ‘The cumulative hierarchy’ and ‘The iterative conception of sets’ are usually taken to be synonymous. However, the second is more general than the first, in that there are recursive procedures that generate some ill-founded sets in addition to well-founded sets. The interesting question is whether or not the arguments in favour of the more restrictive version – the cumulative hierarchy – were all along arguments for the more general version.