Countryman Lines and the Continuum Hypothesis

Abstract

We explore the prospect of basis-like results for the class of Aronszajn lines which are consistent with the Continuum Hypothesis (CH). In particular, we prove that each of the following statements is consistent with CH: Any two Countryman lines contain isomorphic or anti-isomorphic uncountable suborders; for any coherent Aronszajan tree $T\subseteq {}^{<\ddot{I}{‰}_1}\ddot{I}‰$, the filter $\mathcal{U}(T)$ is an ultrafilter. The first result confirms a conjecture of Shelah in the context of CH which was previously shown to follow from the Proper Forcing Axiom. On the other hand, the weak diamond principle $2^{\ddot{I}}‰<2^{\ddot{I}{‰}_1}$ implies that there does not exist a two element basis for the Countryman lines.