Forcing axioms and the complexity of non-stationary ideals
We study the influence of strong forcing axioms on the complexity of the non-stationary ideal on $\omega_2$ and its restrictions to certain cofinalities. Our main result shows that the strengthening $\mathrm{MM}^{++}$of Martin's Maximum does not decide whether the restriction of the non-station...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2022 |
| País: | España |
| Institución: | Universidad de Barcelona |
| Repositorio: | Dipòsit Digital de la UB |
| OAI Identifier: | oai:diposit.ub.edu:2445/217445 |
| Acceso en línea: | https://hdl.handle.net/2445/217445 |
| Access Level: | acceso abierto |
| Palabra clave: | Teoria de conjunts Lògica matemàtica Set theory Mathematical logic |
| Sumario: | We study the influence of strong forcing axioms on the complexity of the non-stationary ideal on $\omega_2$ and its restrictions to certain cofinalities. Our main result shows that the strengthening $\mathrm{MM}^{++}$of Martin's Maximum does not decide whether the restriction of the non-stationary ideal on $\omega_2$ to sets of ordinals of countable cofinality is $\Delta_1$-definable by formulas with parameters in $\mathrm{H}\left(\omega_3\right)$. The techniques developed in the proof of this result also allow us to prove analogous results for the full non-stationary ideal on $\omega_2$ and strong forcing axioms that are compatible with CH. Finally, we answer a question of S . Friedman, Wu and Zdomskyy by showing that the $\Delta_1$-definability of the non-stationary ideal on $\omega_2$ is compatible with arbitrary large values of the continuum function at $\omega_2$. |
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