A Strongly Non-Saturated Aronszajn Tree Without Weak Kurepa Trees
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909661409574912 |
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| author | Krueger, John Stejskalová, Šárka |
| author_facet | Krueger, John Stejskalová, Šárka |
| contents | Assuming the negation of Chang's conjecture, there is a c.c.c. forcing which adds a strongly non-saturated Aronszajn tree. Using a Mahlo cardinal, we construct a model in which there exists a strongly non-saturated Aronszajn tree and the negation of the Kurepa hypothesis is c.c.c. indestructible. For any inaccessible cardinal $κ$, there exists a forcing poset which is Y-proper and $κ$-c.c., collapses $κ$ to become $ω_2$, and adds a strongly non-saturated Aronszajn tree. The quotients of this forcing in intermediate extensions are indestructibly Y-proper on a stationary set with respect to any Y-proper forcing extension. As a consequence, we prove from an inaccessible cardinal that the existence of a strongly non-saturated Aronszajn tree is consistent with the non-existence of a weak Kurepa tree. Finally, we prove from a supercompact cardinal that the existence of a strongly non-saturated Aronszajn tree is consistent with two-cardinal tree properties such as the indestructible guessing model principle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_06878 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Strongly Non-Saturated Aronszajn Tree Without Weak Kurepa Trees Krueger, John Stejskalová, Šárka Logic 03E05, 03E35, 03E40 Assuming the negation of Chang's conjecture, there is a c.c.c. forcing which adds a strongly non-saturated Aronszajn tree. Using a Mahlo cardinal, we construct a model in which there exists a strongly non-saturated Aronszajn tree and the negation of the Kurepa hypothesis is c.c.c. indestructible. For any inaccessible cardinal $κ$, there exists a forcing poset which is Y-proper and $κ$-c.c., collapses $κ$ to become $ω_2$, and adds a strongly non-saturated Aronszajn tree. The quotients of this forcing in intermediate extensions are indestructibly Y-proper on a stationary set with respect to any Y-proper forcing extension. As a consequence, we prove from an inaccessible cardinal that the existence of a strongly non-saturated Aronszajn tree is consistent with the non-existence of a weak Kurepa tree. Finally, we prove from a supercompact cardinal that the existence of a strongly non-saturated Aronszajn tree is consistent with two-cardinal tree properties such as the indestructible guessing model principle. |
| title | A Strongly Non-Saturated Aronszajn Tree Without Weak Kurepa Trees |
| topic | Logic 03E05, 03E35, 03E40 |
| url | https://arxiv.org/abs/2506.06878 |