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Two variants of minimality in forcing extensions

  • Martin Goldstern
  • Tatsuya Goto
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Abstract

We introduce d-minimal and b-minimal extensions, two weakenings of minimality based respectively on eventual domination and infinitely often domination. We prove that the generic extensions obtained by the full Mathias forcing and by Mathias forcing relative to Ramsey ultrafilters are d-minimal, whereas Hechler extensions are b-minimal. We also construct a d-minimal extension that is weakly ωω-bounding but neither minimal nor ωω-bounding, and realize every constellation of these properties compatible with their natural implications.

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Cite

BibTeXRIS
BibTeX
@misc{goldstern2026variants,
  title = {{Two variants of minimality in forcing extensions}},
  author = {Martin Goldstern and Tatsuya Goto},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.17153},
  eprint = {2609.17153},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.17153},
}
RIS
TY  - UNPB
TI  - Two variants of minimality in forcing extensions
AU  - Martin Goldstern
AU  - Tatsuya Goto
PY  - 2026
DA  - 2026/09/15
PB  - arXiv
DO  - 10.48550/arxiv.2609.17153
UR  - https://arxiv.org/abs/2609.17153
AB  - We introduce d-minimal and b-minimal extensions, two weakenings of minimality based respectively on eventual domination and infinitely often domination. We prove that the generic extensions obtained by the full Mathias forcing and by Mathias forcing relative to Ramsey ultrafilters are d-minimal, whereas Hechler extensions are b-minimal. We also construct a d-minimal extension that is weakly ω^ω-bounding but neither minimal nor ω^ω-bounding, and realize every constellation of these properties compatible with their natural implications.
ER  -