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arXiv · 2607.23757

Countryman Lines and the Continuum Hypothesis

John Krueger·Eduardo Martinez Mendoza
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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 $T \subseteq {}^{< \omega_1} \omega$, 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 $2^\omega < 2^{\omega_1}$ implies that there does not exist a two element basis for the Countryman lines.

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John Krueger, Eduardo Martinez Mendoza. 2026-07-26. Countryman Lines and the Continuum Hypothesis. https://arxiv.org/abs/2607.23757

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