论文标题
特殊树号码
The Special Tree Number
论文作者
论文摘要
定义特殊树号，表示为$ \ mathfrak {st} $，是高度$ω_1$的最小尺寸，这既不是特殊也不是Cofinal分支。此红衣主教先前是在$ \ Mathsf {ma} $的片段的背景下进行了研究的，但是在本文中，我们研究了它与其他更典型，更典型的基本特征的关系。古典事实暗示$ \ aleph_1 \ leq \ leq \ mathfrak {st} \ leq 2^{\ aleph_0} $，在马丁的Axiom $ \ Mathfrak {st} = 2^{\ alleph_0} $和该$ \ \ mathfrak {sT} = {sT} = \ alapfrak = aleph_1 = \ alapfrak { $ \ mathsf {ma}（{\ rm纳器}） + 2^{\ aleph_0} =κ$对于任何常规$κ$，因此$ \ mathfrak {st} $的值不是由$ \ mathsf {zfc} $决定的，实际上可能是严格的，并且在本质上可能是基本上是均等的。我们表明，相反，对于任何无法容纳的辅助性的$ \ aleph_0} =κ$，而$ {\ rm non}（\ rm non}（\ mathcal m）= \ mathfrak {a} = \ mathfrak {s s s}尤其是$ \ mathfrak {st} $独立于Cichoń图的Lefthand侧，以及其他方面。证明涉及对标准CCC的深入研究，强迫概念专门化（宽）Aronszajn树，这可能是独立的。
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.
