Title：Three Classes of Logics in the Context of c.c.c. Forcing

Speaker：Han Xiao (University of Hamburg)

Time: 2025/2/18 15:10-18:00

Location: Room 215, Natural Sciences Teaching Building (理教), Peking University

Abstract:

In this talk, I will give a survey of three intermediate logics and their modal counterparts, which play a central role in the study of modal logics of c.c.c. forcing. From a set-theoretic perspective, I will begin with the concept of the destructible gap and, under additional assumptions, construct a new logic that serves as an upper bound in this context, addressing the conjecture proposed by Hamkins, Leibman, and Loewe. Notably, this logic is the least modal companion of Cheq, originally introduced by van Benthem, Guram Bezhanishvili, and Gehrke.

I will present a detailed exposition of these logics, focusing on resolving Nick Bezhanishvili's conjectures on generalized Medvedev logics and exploring the connections between the concept of the nerve in topology and Medvedev logic. This is joint work with Nick Bezhanishvili and Gaëlle Fontaine. Furthermore, I will present results concerning the finite axiomatizability of Cheq, as well as results on the logic of spiked Boolean algebras, a structure proposed by Inamdar and discussed in a paper co-authored with Benedikt Loewe and presented at ICLA.