Definable towers
Tom 256 / 2022
Fundamenta Mathematicae 256 (2022), 221-241
MSC: Primary 03E15.
DOI: 10.4064/fm917-9-2021
Opublikowany online: 18 December 2021
Streszczenie
We study the definability of maximal towers and of inextendible linearly ordered towers (ilt’s), a notion that is more general than that of a maximal tower. We show that there is, in the constructible universe, a $\Pi ^1_1$-definable maximal tower that is indestructible by any proper Suslin poset. We prove that the existence of a $\Sigma ^1_2$ ilt implies that $\omega _1 = \omega _1^L$. Moreover we show that analogous results hold for other combinatorial families of reals. We prove that there is no ilt in Solovay’s model. And finally we show that the existence of a $\Sigma ^1_2$ ilt is equivalent to that of a $\Pi ^1_1$ maximal tower.
