2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/128148We show that it is relatively consistent with ZF that the Borel hierarchy on the reals has length $ω_2$. This implies that $ω_1$ has countable cofinality, so the axiom of choice fails very badly in our model. A similar argument produces models of ZF in which the Borel hierarchy has length any given limit ordinal less than $ω_2$, e.g., $ω$ or $ω_1+ω_1$. Latex2e: 24 pages plus 8 page appendix Latest version at: www.math.wisc.edu/~millerLogic03E15;03E25;03E35text