The hierarchy of ω 1 - Borel sets 1 The hierarchy of ω 1 - Borel sets Arnold

We consider the ω1-Borel subsets of the reals in models of ZFC. This is the smallest family of sets containing the open subsets of the 2 and closed under ω1 intersections and ω1 unions. We show that Martin’s Axiom implies that the hierarchy of ω1-Borel sets has length ω2. We prove that in the Cohen real model the length of this hierarchy is at least ω1 but no more than ω1 + 1. Some authors have considered ω1-Borel sets in other spaces, ω ω1 1 Mekler and Vaananen [10] and or completely metrizable spaces of uncountable density, Willmott [21]. But in this paper we only consider the space 2. Define the levels of the ω1-Borel hierarchy of subsets of 2 ω as follows: 1. Σ0 = Π ∗ 0 = clopen subsets of 2 ω 2. Σα = { ⋃ β<ω1 Aβ : (Aβ : β < ω1) ∈ (Π<α)1} 3. Πα = { ⋃ β<ω1 Aβ : (Aβ : β < ω1) ∈ (Σ<α)1} 4. Π<α = ⋂ β<α Π ∗ β Σ ∗ <α = ⋃ β<α Σ ∗ β The length of this hierarchy is the smallest α ≥ 1 such that Πα = Σ ∗ α. It is easy to show that if α < ω2 and every ω1-Borel set is Π ∗ <α, then Π ∗ β = Σ ∗ β for some β < α, i.e., bounded hierarchies must have a top class (see Miller [11] Proposition 4 p.235). Thanks to the University of Florida Mathematics Department for their support and especially Jindrich Zapletal, William Mitchell, Jean A. Larson, and Douglas Cenzer for inviting me to the special year in Logic 2006-07 during which most of this work was done. Mathematics Subject Classification 2000: 03E15; 03E35; 03E50