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arXiv · 2610.05582

Countryman basis and club-isomorphisms under MA$(ω_1)$

Lucas Polymeris
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Abstract

We show that unlike $\textsf{PFA}$, $\textsf{MA}_{ω_{1}}$ is consistent with the existence of $2^{ω_{1}}$-many Countryman lines which pairwise contain no uncountable isomorphic or reverse-isomorphic suborders. This clarifies an issue regarding the claim that $\textsf{MA}_{ω_{1}}$ implies that there is a finite basis for the Countryman lines. The family of Countryman lines comes from a family of $2^{ω_{1}}$-many coherent trees which pairwise contain no club-isomorphic subtrees. To prove this, we first construct such a family of trees using $\diamondsuit$, and then we iterate ccc forcings while preserving this property.

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Lucas Polymeris. 2026-10-04. Countryman basis and club-isomorphisms under MA$(ω_1)$. https://arxiv.org/abs/2610.05582

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