Cores of Π₁¹ sets of reals

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 (Π11) and Borel (Δ11) with their lightface analogs, would represent every Π11 subset of the real line as the union of a chain of Δ11 sets. No such analog is true, however, because some Δ11 sets are not the union of their Δ11 subsets. For example, the set W, consisting of those reals which code well-orderings (in some standard coding) is Π11, but, by the boundedness principle ([3], [9]), any Δ11 subset of W contains codes only for well-orderings shorter than ω1, the first nonrecursive ordinal. Accordingly, we define the core of a Π11 set to be the union of its Δ11 subsets; clearly this is the largest subset of the given Π11 set for which an effective version of the classical representation could exist.In §1, we develop the elementary properties of cores of Π11 sets. For example, such a core is itself Π11 and can be represented as the union of a chain of Δ11 sets in a natural way; the chain will have length at most ω1. We show that the core of a Π11 set is “almost all” of the set, while on the other hand there are uncountable Π11 sets with empty cores.

How to Compute Antiderivatives

“…Ajtai's result is unpublished but it is referenced in an article by Dougherty and Kechris [8], who outline their own proof of this result. They call this type of definability "∆ 1 1 on the set of derivative codes". But their main thrust was to show that no substantial improvement to this will ever be possible for any solution to the primitive problem.…”

“…Harel and Kozen prove that the relations which can be programmed in an "inductive" way as defined above, are exactly the ones which can be defined using a transfinite induction G = S G α where at limit ordinals we take unions and where G α+1 is defined "arithmetically" and "positively" from G α as in the previous example. (Work of Kleene classifies these inductive sets as the "Π 1 1 sets", i.e., sets which can be defined using an arithmetical formula preceded by one "8" quantifier over infinite sequences of integers, so that the hyperarithmetical sets are the same as the "∆ 1 1 " sets, i.e., sets which are both Π 1 1 and whose complements are Π 1 1 .) It has long been known that the classes of "inductive" and "hyperarithmetical" sets reflect this kind of algorithmic behavior.…”

“…Then running the programs simultaneously, (see [9] for technical details of how to create a single IND program to combine two others) we have a computation which will halt on all derivatives. This level of complexity is what Dougherty and Kechris call "∆ 1 1 on the set of [codes of] derivatives" since although it is not strictly "hyperarithmetical", it always halts as long as we are given the code of a derivative. In other words, since the primitive problem says "Given a derivative .…”