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PreprintCC BY 4.0arXiv

Countryman basis and club-isomorphisms under MA(ω1)

  • Lucas Polymeris
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Abstract

We show that unlike textsfPFA, textsfMAω1 is consistent with the existence of 2ω1-many Countryman lines which pairwise contain no uncountable isomorphic or reverse-isomorphic suborders. This clarifies an issue regarding the claim that textsfMAω1 implies that there is a finite basis for the Countryman lines. The family of Countryman lines comes from a family of 2ω1-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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Cite

BibTeXRIS
BibTeX
@misc{polymeris2026countryman,
  title = {{Countryman basis and club-isomorphisms under MA$(ω_1)$}},
  author = {Lucas Polymeris},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.05582},
  eprint = {2610.05582},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.05582},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Countryman basis and club-isomorphisms under MA(ω_1)
AU  - Lucas Polymeris
PY  - 2026
DA  - 2026/10/04
PB  - arXiv
DO  - 10.48550/arxiv.2610.05582
UR  - https://arxiv.org/abs/2610.05582
AB  - We show that unlike textsfPFA, textsfMA_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 textsfMA_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.
ER  -