  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
