Events

Title: Dependence logic and its axiomatization problem
Speaker's homepage

Dependence logic, introduced by Väänänen (2007), is a non-classical logic for reasoning about dependence and independence. The logic extends first-order logic with a new type of atomic formulas, called dependence atoms, to specify explicitly the dependence relation between variables. Dependence logic adopts an innovative semantics, called team semantics (Hodges 1997), in which formulas are evaluated on a model with respect to sets of assignments (called teams), instead of single assignments. Teams are essentially relations on the model. For this reason, dependence logic is equi-expressive with existential second-order logic, and thus not fully axiomatizable. In this talk, I will give a concise introduction to dependence logic, and I will also survey recent developments in finding partial axiomatizations for the logic.

Title: Ccc without C, si? Si.
Speaker's homepage

What does the countable chain condition mean without the axiom of choice? We will discuss several possible definitions, all equivalent in ZFC, none equivalent in ZF(+DC). We will also present two "external" definitions (due to Bukovský and to Mekler) and see how they fit into this picture.

We will show that a ccc forcing can collapse ω1, and quite possibly be countably closed while doing so. On the other hand, with the "correct definition" of ccc, no cofinalities or cardinals are changed above ω1. Whether or not ω1can be collapsed is open, but we know that would require it to be singular.

This is a joint work with Noah Schweber.