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Definable towers

Volume 256 / 2022

Vera Fischer, Jonathan Schilhan Fundamenta Mathematicae 256 (2022), 221-241 MSC: Primary 03E15. DOI: 10.4064/fm917-9-2021 Published online: 18 December 2021


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.


  • Vera FischerInstitute of Mathematics
    University of Vienna
    Kolingasse 14-16
    1090 Wien, Austria
  • Jonathan SchilhanSchool of Mathematics
    University of East Anglia
    Norwich Research Park
    Norwich NR4 7TJ, UK

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