On the complexity of classes of uncountable structures: trees on $\aleph _1$
| Autor: | Dániel T. Soukup, Sy-David Friedman |
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| Rok vydání: | 2021 |
| Předmět: | |
| Zdroj: | Fundamenta Mathematicae. 253:175-196 |
| ISSN: | 1730-6329 0016-2736 |
| Popis: | We analyse the complexity of the class of (special) Aronszajn, Suslin and Kurepa trees in the projective hierarchy of the higher Baire-space $\omega_1^{\omega_1}$. First, we will show that none of these classes have the Baire property (unless they are empty). Moreover, under $(V=L)$, (a) the class of Aronszajn and Suslin trees is $\Pi_1^1$-complete, (b) the class of special Aronszajn trees is $\Sigma_1^1$-complete, and (c) the class of Kurepa trees is $\Pi^1_2$-complete. We achieve these results by finding nicely definable reductions that map subsets $X$ of $\omega_1$ to trees $T_X$ so that $T_X$ is in a given tree-class $\mathcal T$ if and only if $X$ is stationary/non-stationary (depending on the class $\mathcal T$). Finally, we present models of CH where these classes have lower projective complexity. Comment: 16 pages |
| Databáze: | OpenAIRE |
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