The special tree number
Volume 262 / 2023
Abstract
Define the special tree number, denoted $\mathfrak s\mathfrak t$, 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 $\aleph _1 \leq \mathfrak s\mathfrak t \leq 2^{\aleph _0}$, under Martin’s Axiom $\mathfrak s\mathfrak t = 2^{\aleph _0}$, and that $\mathfrak s\mathfrak t = \aleph _1$ 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 s\mathfrak t = 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, $\mathfrak s\mathfrak t$ is independent of the left hand 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.
