Images of Borel sets and k-analytic sets

Borel Sets and Countable Models

We show that certain families of sets and functions related to a countable structure A are analytic subsets of a Polish space. Examples include sets of automorphisms, endomorphisms and congruences of A and sets of the combinatorial nature such as coloring of countable plain graphs and domino tiling of the plane. This implies, without any additional set-theoretical assumptions, i.e., in ZFC alon...

Events of Borel Sets, Construction of Borel Sets and Random Variables for Stochastic Finance

We consider special events of Borel sets with the aim to prove, that the set of the irrational numbers is an event of the Borel sets. The set of the natural numbers, the set of the integer numbers and the set of the rational numbers are countable, so we can use the literature [10] (pp. 78-81) as a basis for the similar construction of the proof. Next we prove, that different sets can construct ...

A Course on 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...

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...