Long Borel hierarchies

We show that there is a model of ZF in which the Borel hierarchy on the reals has length 2. This implies that 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 exactly +1 levels for any given limit ordinal less than 2. We also show that assuming a large cardinal hypothesis there are models of ZF in which the Borel hierarchy is arbitrarily long. Contents

• Type: Journal
• Year: 2008
• Where: MLQ
• Authors: Arnold W. Miller
• Keywords: Borel Hierarchy, Countable Cofinality, MLQ 2008, Model