Representing projective sets as unions of Borel sets

Representing Projective Sets as Unions of Borel Sets

We consider a method of representing projective sets by a particular type of union of Borel sets, assuming AD. We prove a generalization of the theorem that a set is S^ iff it is the union of a>x Borel sets. The Axiom of Determinacy (AD) is always assumed. Theorem 1 (Sierpinski, Solovay, Moschovakis [8, 7D.10]). Let A c com. A is Ej iff A is the union of cox Borel sets. The purpose of this pape...

On Discrete Borel Spaces and Projective Sets

Let J denote the unit interval, S—IXI the unit square; Cj and Cs the class of all subsets of I and 6, respectively. By Cj X Cj is meant the or-algebra on S generated by rectangles with sides in C/. The purpose of this note is to prove the following theorem (which settles a problem of S. M. Ulam) and observe some of its consequences. Without explicit mention, the axiom of choice has been assumed...

Finite Sets as Complements of Finite Unions of Convex Sets

Suppose S ⊆ R is a set of (finite) cardinality n whose complement can be written as the union of k convex sets. It is perhaps intuitively appealing that when n is large k must also be large. This is true, as is shown here. First the case in which the convex sets must also be open is considered, and in this case a family of examples yields an upper bound, while a simple application of a theorem ...

A Course on Borel Sets

Borel Sets with Large Squares

For a cardinal μ we give a sufficient condition ⊕μ (involving ranks measuring existence of independent sets) for: ⊗μ: if a Borel set B ⊆ R × R contains a μ-square (i.e. a set of the form A × A, |A| = μ) then it contains a 20 -square and even a perfect square. And also for ⊗′μ: if ψ ∈ Lω1,ω has a model of cardinality μ then it has a model of cardinality continuum generated in a “nice”, “absolute...