A note on hierarchies of Borel type sets

A Course on Borel Sets

Long Borel hierarchies

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.

On monotone presentations of Borel sets

If A is a Σξ set and An (n < ω) are Borel sets then we call {An : n < ω} a presentation of A if A = S n<ω An and An (n < ω) have lower Borel class than A has. We show that for 2 ≤ ξ < ω1 it is not possible to assign a presentation to Σξ sets in a monotone way, i.e., it is not possible to define functions fn : Σ 0 ξ → Πξ (n < ω) such that for every A ∈ Σξ we have A = S n<ω fn(A) and A,A ′ ∈ Σξ, ...

On open maps of Borel sets

We answer in the affirmative [Th. 3 or Corollary 1] the question of L. V. Keldysh [5, p. 648]: can every Borel set X lying in the space of irrational numbers P not Gδ · Fσ and of the second category in itself be mapped onto an arbitrary analytic set Y ⊂ P of the second category in itself by an open map? Note that under a space of the second category in itself Keldysh understood a Baire space. T...

A note on hierarchies and bureaucracies

In this note, we argue that there is a bug in [Tirole, J., “Hierarchies and bureaucracies: On the role of collusion in organizations,” Journal of Law, Economics and Organization, vol.2, 181-214, 1986].