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Hagendorf Orders

  • Jonathan Schilhan
  • Thilo Weinert
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Abstract

Call a linear order type φ Hagendorf if it shares two properties with additively indecomposable ordinal numbers without being one itself: φ is strictly indecomposable to the right and whenever ψ< φ, then ψ can be embedded into a proper initial segment of φ. In the 1970s, J. Hagendorf asked whether such types exist. F. Galvin observed that they must be uncountable, and soon thereafter, J. Larson proved that they cannot be scattered. We provide a simplification of her argument that might give additional insight into the σ-scattered case. We then show that consistently Hagendorf types exist and that this holds in many models of set theory, for instance under BA or MAℵ1. In fact, their non-existence has large cardinal strength. Furthermore, we construct real Hagendorf types from diamondsuit and from PFA. This marks the first progress on this interesting problem since its description by Larson four dozen years ago.

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Cite

BibTeXRIS
BibTeX
@misc{schilhan2026hagendorf,
  title = {{Hagendorf Orders}},
  author = {Jonathan Schilhan and Thilo Weinert},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.39487},
  eprint = {2609.39487},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.39487},
}
RIS
TY  - UNPB
TI  - Hagendorf Orders
AU  - Jonathan Schilhan
AU  - Thilo Weinert
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.39487
UR  - https://arxiv.org/abs/2609.39487
AB  - Call a linear order type φ Hagendorf if it shares two properties with additively indecomposable ordinal numbers without being one itself: φ is strictly indecomposable to the right and whenever ψ< φ, then ψ can be embedded into a proper initial segment of φ. In the 1970s, J. Hagendorf asked whether such types exist. F. Galvin observed that they must be uncountable, and soon thereafter, J. Larson proved that they cannot be scattered. We provide a simplification of her argument that might give additional insight into the σ-scattered case. We then show that consistently Hagendorf types exist and that this holds in many models of set theory, for instance under BA or MA_ℵ_1. In fact, their non-existence has large cardinal strength. Furthermore, we construct real Hagendorf types from diamondsuit and from PFA. This marks the first progress on this interesting problem since its description by Larson four dozen years ago.
ER  -