Comments on Choiceless Chain Conditions

Abstract

In [10] it was suggested that the countable chain condition should be defined in the absence of choice as "every predense set contains a countable predense subset". During his tutorial at the "120 Years of Choice" conference in 2024, Philipp Schlicht remarked that this definition does not have an iteration theorem in $\mathsf{ZF}$. We provide a proof of this claim. Specifically, we show that if every finite support iteration of ccc forcings is again ccc, then the Axiom of Choice for countable families of countable sets must hold. However, we prove that under the assumption of the Principle of Dependent Choice, the finite support iteration of ccc forcings is again ccc. This requires correcting some issues with the definition of properness (and therefore Mekler's definition of ccc) from [1]. We then study Knaster principles and their relations to this version of ccc.