Logic Seminar: Some fruit from set theoretic pluralism Part 2

Abstract: Unlike arithmetic, the identification of a canonical model of set theory is incredibly difficult. In this respect, the situation is more akin to geometry than arithmetic. We are capable of simulating alternative models with different structural properties within V, using either boolean valued forcing or a simulated ultrapower. This is similar to the simulation of projective and spherical geometries in euclidean space. This begs the question “is there actually a canonical model of ZFC, as in the case of PA, or is this indication that there is an interesting plethora of models of set theory, similar to the plethora of geometries?”

Both positions, called the universe and multiverse views, have their advocates, who, to find evidence for their positions, have produced a large amount of deep and interesting mathematics. In this series of talks, we will look at some of the mathematics that has come out of the multiverse camp. Topics include a natural model for the multiverse axioms, the modal logic of forcing, and set theoretic geology.