2007/04/30 by Arnold W. Miller, Miller, Arnold W.
Computer Science · Economics, Econometrics and Finance · Mathematics · #03E15 #03E25 #03E35 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #Economic theories and models #FOS: Mathematics #Logic (math.LO) #math.LO #msc:03E15 #msc:03E25 #msc:03E35
paper · pdf · doi:10.48550/arxiv.0704.3998
arxiv created 2007/04/30 · openalex publication_date 2007/04/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that it is relatively consistent with ZF that the Borel hierarchy on the reals has length ω2. This implies that ω1 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 ω2, e.g., ω or ω1+ω1. Latex2e: 24 pages plus 8 page appendix Latest version at: www.math.wisc.edu/~miller