Title: Long Borel Hierarchies
Author: Arnold W. Miller
Abstract: 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
There is an on-line only appendix.