Definable Towers

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 ${\mathrm{\Pi }}_1^1$ definable maximal tower that is indestructible by any proper Suslin poset. We prove that the existence of a ${\mathrm{\Sigma }}_2^1$ 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 ${\mathrm{\Sigma }}_2^1$ ilt is equivalent to that of a ${\mathrm{\Pi }}_1^1$ maximal tower.