Logic Seminar: Session 2
Some have seen the potential infinite as offering a cure-all to the problems of contemporary philosophy of mathematics. But, as with all such general solutions, it must be asked how much mathematics can one recover in the setting of the potential infinite? In this talk we turn our attention to Frege’s theorem. This is for three reasons. First, the theorem is known to be robust in the face of change of logic: it continues to hold in the intuitionist setting and with some limited comprehension. Second, the potential infinite seems almost uniquely suited to help the neo-logicist respond to worries about logicism being committed to an actual infinity of objects. Third, Hodes famously tried to pursue a modal version of logicism but the status of Frege’s theorem is not settled by his work. In particular, Hodes' claimed this version of logicism ran into problems due to the lack of cross-world predication. But in the intervening years since Hodes’ work, there has been both great advances in modal logic which allow for more subtle forms of cross-world predication, and sophisticated work on modal versions of other foundational theories, primarily set theory. In this talk, we show that certain forms of the modal analogue of Frege’s theorem fail. But importantly, Hodes misidentified the problem: the problem is not with cross-world predication but rather with the interpretation of comprehension.
Chair for Session: tba