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arXiv · 0704.3998

Long Borel Hierarchies

Arnold W. Miller
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Abstract

We show that it is relatively consistent with ZF that the Borel hierarchy on the reals has length $ω_2$. This implies that $ω_1$ 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 $ω_2$, e.g., $ω$ or $ω_1+ω_1$. Latex2e: 24 pages plus 8 page appendix Latest version at: www.math.wisc.edu/~miller

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Arnold W. Miller. 2007-04-30. Long Borel Hierarchies. https://arxiv.org/abs/0704.3998

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