The Special Tree Number

Abstract

Define the special tree number, denoted $\mathfrak{st}$, to be the least size of a tree of height $ω_1$ which is neither special nor has a cofinal branch. This cardinal had previously been studied in the context of fragments of $\mathsf{MA}$ but in this paper we look at its relation to other, more typical, cardinal characteristics. Classical facts imply that $\aleph_1 \leq \mathfrak{st} \leq 2^{\aleph_0}$, under Martin's Axiom $\mathfrak{st} = 2^{\aleph_0}$ and that $\mathfrak{st} = \aleph_1$ is consistent with $\mathsf{MA}({\rm Knaster}) + 2^{\aleph_0} = κ$ for any regular $κ$ thus the value of $\mathfrak{st}$ is not decided by $\mathsf{ZFC}$ and in fact can be strictly below essentially all well studied cardinal characteristics. We show that conversely it is consistent that $\mathfrak{st} = 2^{\aleph_0} = κ$ for any $κ$ of uncountable cofinality while ${\rm non}(\mathcal M) = \mathfrak{a} = \mathfrak{s} = \mathfrak{g} = \aleph_1$. In particular $\mathfrak{st}$ is independent of the lefthand side of Cichoń's diagram, amongst other things. The proof involves an in depth study of the standard ccc forcing notion to specialize (wide) Aronszajn trees, which may be of independent interest.