This paper is concerned with the hyderdegrees of elements of uncountable Borel subsets of ωω. The Borel subsets of ωω are the so-called Δ11 subsets of ωω, which are the subsets of ωω that are Δ11 in some parameter f: ω → ω.

The results of this paper were inspired by two earlier results about the hyperdegrees of elements of Σ11 subsets of ωω. 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 Σ11 subsets of ωω (see Rogers [3, p. 421]). The second is that there exists an uncountable Σ11 subset of ωω 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 Σ11 by Δ11 in Gandy's result? Can we replace Σ11 by Δ11 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 Δ11 subsets of ωω with no hyperarithmetic elements. We conjecture that this holds also for uncountable Δ11 subsets of ωω. This conjecture is open despite the recent claim in Notices of the American Mathematical Society, vol. 19 (1972), p. A616, Abstract 696-02-10.