The Special Tree Number

TL;DR

This paper introduces the special tree number, a cardinal characteristic related to trees of height ω₁, and explores its independence and relationships with other cardinal invariants through forcing techniques.

Contribution

It establishes the independence of the special tree number from many well-known cardinal characteristics and analyzes its relation to other invariants using ccc forcing methods.

Findings

The special tree number can be equal to 2^{ℵ₀} for any uncountable cofinality κ.

It is consistent that the special tree number is strictly below many classical invariants.

The paper provides an in-depth study of ccc forcing to specialize Aronszajn trees.

Abstract

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.