Sets of ordinals constructible from trees and the third Victoria Delfino problem

Abstract

A very important part of the structure theory of Σ^1_2 sets of reals is based on their close interrelationship with the Gödel constructible universe L. The fundamental fact underlying this connection is the theorem of Shoenfield which asserts that every Σ^1_2 set of reals is Souslin over L. This means that given any Σ^1_2 subset of the reals (=ω^ω in this paper), there is a tree T on ω x λ (λ some ordinal, which can be taken to be ℵ_l here) such that T є L and A = p[T] = {ɑ є ω^ω: : ∃f є λ^ω ∀n(ɑ↾n,f↾n) є T}.

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