Omitting types: application to descriptive set theory

Omitting Types: Application to Descriptive Set Theory

The omitting types theorem of infinitary logic is used to prove that every small II set of analysis or any small 2. set of set theory is constructible. In what follows we could use either the omitting types theorem for infinitary logic or the same theorem for what Grilliot[2] calls (eA)-logic. I find the latter more appealing. Suppose i_ is a finitary logical language containing the symbols of ...

Descriptive Set Theory Descriptive Set Theory

Introduction to Descriptive Set Theory

Mathematicians in the early 20th century discovered that the Axiom of Choice implied the existence of pathological subsets of the real line lacking desirable regularity properties (for example nonmeasurable sets). This gave rise to descriptive set theory, a systematic study of classes of sets where these pathologies can be avoided, including, in particular, the definable sets. In the first half...

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)...

Some descriptive set theory

Definition 1.1. Let (X, τ) be a topological space. A subset D ⊆ X is called dense if D ∩O 6= ∅ for every nonempty open set O ⊆ X. X is called separable if X has a countable dense subset. X is called metrizable if there is a metric d on X such that the topology τ is the same as the topology induced by the metric. The metric is called complete if every Cauchy sequence converges in X. Finally, X i...