Mathematical Logic Quarterly 54 (3):307-322 (2008)

We show that there is a model of ZF in which 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 exactly λ + 1 levels for any given limit ordinal λ less than ω2. We also show that assuming a large cardinal hypothesis there are models of ZF in which the Borel hierarchy is arbitrarily long
Keywords Axiom of choice  Borel hierarchies  countable unions
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DOI 10.1002/malq.200710044
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References found in this work BETA

Zur Axiomatik der Mengenlehre (Fundierungs‐ und Auswahlaxiom).Ernst Specker - 1957 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 3 (13‐20):173-210.
The Axiom of Choice.Thomas J. Jech - 1973 - Amsterdam, Netherlands: North-Holland.
Zur Axiomatik der Mengenlehre.Ernst Specker - 1957 - Mathematical Logic Quarterly 3 (13-20):173-210.
Successive Weakly Compact or Singular Cardinals.Ralf-Dieter Schindler - 1999 - Journal of Symbolic Logic 64 (1):139-146.

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