Modal set theory *

Set theory is the study of sets using the tools of contemporary mathematical logic. Modal set theory draws in particular upon contemporary modal logic, the logic of necessity and possibility. One simple and obvious motivation for modal set theory is the fact that, from a realist perspective that takes the existence of sets seriously, sets have philosophically interesting modal properties. For instance, perhaps the most notable and distinctive property of sets is their extensionality: sets a and b are identical if they have exactly the same members; formally, where we take variables from the lower end of the alphabet to range over sets: