Long Borel Hierarchies

TL;DR

The paper demonstrates the relative consistency of the Borel hierarchy on reals having length 2 within ZF set theory, showing that 1 can have countable cofinality and that the hierarchy length can be any limit ordinal less than 2.

Contribution

It constructs models of ZF where the Borel hierarchy reaches various specified lengths, illustrating the hierarchy's potential complexity without the axiom of choice.

Findings

Borel hierarchy can have length 2 in ZF models.

1 can have countable cofinality in these models.

Hierarchy length can be any limit ordinal less than 2.

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