The Special Tree Number

Corey Bacal Switzer

Published 2022-03-08Version 1

We introduce a new cardinal characteristic called the {\em special tree number}, denoted $\mathfrak{st}$, defined to be the least size of a tree of height $\omega_1$ which is neither special nor has a cofinal branch. Classical facts imply that $\aleph_1 \leq \mathfrak{st} \leq 2^{\aleph_0}$, $\mathsf{MA}$ implies $\mathfrak{st} = 2^{\aleph_0}$ while $\mathfrak{st} = \aleph_1$ 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 regular $\kappa$ 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.