The club principle and the distributivity number

Abstract

We give an affirmative answer to Brendle's and Hrušák's question of whether the club principle together with 𝔥 > ℵ₁ is consistent. We work with a class of axiom A forcings with countable conditions such that q≥_n p is determined by finitely many elements in the conditions p and q and that all strengthenings of a condition are subsets, and replace many names by actual sets. There are two types of technique: one for tree-like forcings and one for forcings with creatures that are translated into trees. Both lead to new models of the club principle.