Long Borel Hierarchies
A.Miller
1
Long Borel Hierarchies
Long Borel Hierarchies
arXiv:0704.3998v1 [math.LO] 30 Apr 2007
Arnold W. Miller
1
Abstract 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 . Introduction In this paper we do not assume the axiom of choice, not even in the form of choice functions for countable families. Define the classical Borel families, Π0α and Σ0α , of subsets of 2ω for any ordinal α as usual: 1. Σ00 = Π00 =clopen subsets of 2ω , 2. Π0
1
Long Borel Hierarchies
Long Borel Hierarchies
arXiv:0704.3998v1 [math.LO] 30 Apr 2007
Arnold W. Miller
1
Abstract 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 . Introduction In this paper we do not assume the axiom of choice, not even in the form of choice functions for countable families. Define the classical Borel families, Π0α and Σ0α , of subsets of 2ω for any ordinal α as usual: 1. Σ00 = Π00 =clopen subsets of 2ω , 2. Π0
