Define the special tree number, denoted $\mathfrak{ st}$, to be the least size of a tree of height $\omega_1$ 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 $\mathsf{ MA}({ \rm Knaster}) + 2^{ \aleph_0} = \kappa$ for any regular $\kappa$ 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 $\mathfrak{ st} = 2^{ \aleph_0} = \kappa$ for any $\kappa$ 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 Cicho\'{ n}'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.
Categories: math.LO