Maximal almost disjoint families, determinacy, and forcing
| Autor: | Haga, Karen Bakke, Schrittesser, David, Törnquist, Asger |
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| Rok vydání: | 2018 |
| Předmět: | |
| Zdroj: | Journal of Mathematical Logic Vol. 22, No. 01, 2150026 (2022) |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1142/S0219061321500264 |
| Popis: | We study the notion of $\mathcal J$-MAD families where $\mathcal J$ is a Borel ideal on $\omega$. We show that if $\mathcal J$ is an arbitrary $F_\sigma$ ideal, or is any finite or countably iterated Fubini product of $F_\sigma$ ideals, then there are no analytic infinite $\mathcal J$-MAD families, and assuming Projective Determinacy there are no infinite projective $\mathcal J$-MAD families; and under the full Axiom of Determinacy + $V=\mathbf{L}(\mathbb{R})$ there are no infinite $\mathcal J$-mad families. These results apply in particular when $\mathcal J$ is the ideal of finite sets $\mathrm{Fin}$, which corresponds to the classical notion of MAD families. The proofs combine ideas from invariant descriptive set theory and forcing. Comment: 40 pages |
| Databáze: | arXiv |
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