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Asaf Karagila (University of East Anglia)

Category
Models and Sets Seminar
Date
Date
Tuesday 22 March 2022
Category

Ccc without C, si? Si.

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.