Cores of Π11 sets of reals

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Abstract
A classical result of descriptive set theory expresses every co-analytic subset of the real line as the union of an increasing sequence of Borel sets, the length of the chain being at most the first uncountable ordinal ℵ1 (see [5], [8]). An effective analog of this theorem, obtained by replacing co-analytic (Π1 1) and Borel (Δ1 1) with their lightface analogs, would represent every Π1 1 subset of the real line as the union of a chain of Δ1 1 sets. No such analog is true, however, because some Δ1 1 sets are not the union of their Δ1 1 subsets. For example, the set W, consisting of those reals which code well-orderings (in some standard coding) is Π1 1, but, by the boundedness principle ([3], [9]), any Δ1 1 subset of W contains codes only for well-orderings shorter than ω 1, the first nonrecursive ordinal. Accordingly, we define the core of a Π1 1 set to be the union of its Δ1 1 subsets; clearly this is the largest subset of the given Π1 1 set for which an effective version of the classical representation could exist. In §1, we develop the elementary properties of cores of Π1 1 sets. For example, such a core is itself Π1 1 and can be represented as the union of a chain of Δ1 1 sets in a natural way; the chain will have length at most ω 1. We show that the core of a Π1 1 set is “almost all” of the set, while on the other hand there are uncountable Π1 1 sets with empty cores.

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