Borel Sets and Hyperdegrees

This paper is concerned with the hyderdegrees of elements of uncountable Borel subsets of ω ω . The Borel subsets of ω ω are the so-called Δ 1 1 subsets of ω ω , which are the subsets of ω ω that are Δ 1 1 in some parameter f : ω → ω . The results of this paper were inspired by two earlier results about the hyperdegrees of elements of Σ 1 1 subsets of ω ω . The first is known as the Gandy basis theorem, and asserts that the collection of functions of strictly lower hyperdegree than the hyperjump form a basis for the Σ 1 1 subsets of ω ω (see Rogers [3, p. 421]). The second is that there exists an uncountable Σ 1 1 subset of ω ω no element of which is hyperarithmetic in any other. The latter is credited to Feferman in Harrison [1], and appears there as Corollary 2.7, p. 537. Two questions arise. Can we find other basis theorems if we replace Σ 1 1 by Δ 1 1 in Gandy's result? Can we replace Σ 1 1 by Δ 1 1 in Feferman's result? We simultaneously answer the first positively, and the second negatively by proving that any hyperdegree, in which the hyperjump is hyperarithmetic, forms a basis for the Δ 1 1 subsets of ω ω with no hyperarithmetic elements. We conjecture that this holds also for uncountable Δ 1 1 subsets of ω ω . This conjecture is open despite the recent claim in Notices of the American Mathematical Society, vol. 19 (1972), p. A616, Abstract 696-02-10.