Long Borel Hierarchies

We show that it is relatively consistent with ZF that the Borel hierarchy on the reals has length $\omega_2$. This implies that $\omega_1$ 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 $\omega_2$, e.g., $\omega$ or $\omega_1+\omega_1$. Latex2e: 24 pages plus 8 page appendix Latest version at: www.math.wisc.edu/~miller