Specialising Aronszajn trees by countable approximations (Q1411659)

The main point of the paper is finding forcing notions specializing an Aronszajn tree, which are creature forcing, tree-like with halving, but being based on~\(\omega_1\) rather than on~\(\omega\). To specialize an Aronszajn tree it is necessary to add a~subset of~\(\omega_1\) and so the creature forcing introduced by \textit{A. Rosłanowski} and \textit{S. Shelah} [Norms on possibilities. I: Forcing with trees and creatures, Mem. Am. Math. Soc. 671 (1999; Zbl 0940.03059)] cannot be applied. The authors modify some assumption on countability in the definition of creatures and use these creatures in the definition of the forcing. Conditions of the forcing are functions~\(p\) such that the range of~\(p\) is a~subset of~\(\omega\) and the domain of~\(p\) is an \(\omega\)-tree of partial specializing functions on a~given Aronszajn tree~\(\mathbf T\). The immediate successors of a~condition are described by certain creatures. The constructed forcing notion is proper, \(^\omega\omega\)-bounding, and adds a~function which specializes the tree~\(\mathbf T\). Consequently, the countable support iteration of these forcing notions over the model with \(2^{\aleph_1}=\aleph_2\) produces a~model in which all Aronszajn trees are special, \(\mathfrak b=\omega_1\), and \(2^\omega=\aleph_2\). In fact, the work grew from attempts of the authors to show the consistency of~SH together with~\(\clubsuit\).