Logic Seminar: Session 8
One of the strongest arguments for forcing axioms as an extension of our best axiomatization of set theory, put forth prominently by Stevo Todorcevic, is their ability to provide a satisfactory structure theory for P(w1), much like how large cardinal axioms provide a satisfactory structure theory for P(w). In this way, the case for forcing axioms closely resembles at least part of the widely accepted argumentation for large cardinal axioms. Opponents of forcing axioms, such as Peter Koellner, have responded to this argument by proposing that the structure theory for P(w1) can be preserved by relegating it to an inner model of the set-theoretic universe, thereby allowing alternative axioms to hold at the level of V. In this talk, I will present both Todorcevic’s argument for forcing axioms as well as Koellner’s response, focusing on the similarities and differences between the case for forcing axioms and past instances of relegating axioms to inner models: in particular, I will focus on the case of determinacy. I will conclude by noting some challenges for both sides in this dispute, as well as highlighting important questions whose resolution might aid in deciding this question more conclusively.
Chair for Session: tba