Long Borel Hierarchies

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