Descriptive set Theory of Families of Small Sets

Descriptive Set Theory of Families of Small Sets

This is a survey paper on the descriptive set theory of hereditary families of closed sets in Polish spaces. Most of the paper is devoted to ideals and σ-ideals of closed or compact sets.

Descriptive Set Theory Descriptive Set Theory

Descriptive Set Theory : Projective Sets , in Handbook of

Vienna notes on effective descriptive set theory and admissible sets

These notes are based around the exposition of a sequence of connected ideas: ∆1 and Π 1 1 and Σ 1 1 sets; admissible structures; ω 1 ; recursive well founded relations; ordinal analysis of trees; Barwise compactness; Gandy-Harrington forcing. I am not going to discuss applications of these techniques – though there are striking applications such as Harrington’s proof of Silver’s theorem, or th...

Descriptive Set Theory Problem Set

Prove that any strictly monotone sequence (Uα)α<γ of open subsets of X has countable length, i.e. γ is countable. Hint: Use the same idea as in the proof of (a). (c) Show that every monotone sequence (Uα)α<ω1 open subsets of X eventually stabilizes, i.e. there is γ < ω1 such that for all α < ω1 with α ≥ γ, we have Uα = Uγ. Hint: Use the regularity of ω1. (d) Conclude that parts (a), (b) and (c)...