The failure of square at all uncountable cardinals is weaker than a Woodin limit of Woodin cardinals
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866915796983218176 |
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| author | Blue, Douglas Larson, Paul Sargsyan, Grigor |
| author_facet | Blue, Douglas Larson, Paul Sargsyan, Grigor |
| contents | We force the Axiom of Choice over the least initial segment of a Nairian model satisfying ZF. In the forcing extension, square_kappa fails at all uncountable cardinals kappa, and every regular cardinal is omega-strongly measurable in HOD, as witnessed by the omega-club filter. Thus the failure of square everywhere is within the current reach of inner model theory, and the HOD Hypothesis is not provable in ZFC. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_13077 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The failure of square at all uncountable cardinals is weaker than a Woodin limit of Woodin cardinals Blue, Douglas Larson, Paul Sargsyan, Grigor Logic 03E45 We force the Axiom of Choice over the least initial segment of a Nairian model satisfying ZF. In the forcing extension, square_kappa fails at all uncountable cardinals kappa, and every regular cardinal is omega-strongly measurable in HOD, as witnessed by the omega-club filter. Thus the failure of square everywhere is within the current reach of inner model theory, and the HOD Hypothesis is not provable in ZFC. |
| title | The failure of square at all uncountable cardinals is weaker than a Woodin limit of Woodin cardinals |
| topic | Logic 03E45 |
| url | https://arxiv.org/abs/2602.13077 |